Nguyên hàm và các phương pháp tìm nguyên hàm – Trần Văn Tài Toán 12

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TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 1 | THBTN
ⓣⓗ

1. 

()fx

.K

()Fx


()fx
trên
K

( ) ( ), .F x f x x K

()Fx

()fx
trên
K

()fx
trên
K

( ) ( ) , .f x dx F x C const C
2. 

K
và
0k
thì ta luôn có:
( ) ( ) .f x dx f x C
( ) ( ) .kf x dx k f x dx
( ) ( ) ( ) ( )f x g x dx f x dx g x dx
C 
1
1
x
x dx C
1
1 ( )
()
1
n
n
ax b
ax b dx C
an
1
lndx x C
x
11
lndx ax b C
ax b a
2
11
dx C
x
x
2
1 1 1
()
dx C
a ax b
ax b
sin cosx dx x C
1
sin( ) cos( )ax b dx ax b C
a
cos sinx dx x C
1
cos( ) sin( )ax b dx ax b C
a
2
1
cot
sin
dx x C
x
2
11
cot( )
sin ( )
dx ax b C
a
ax b
2
1
tan
cos
dx x C
x
2
11
tan( )
cos ( )
dx ax b C
a
ax b
xx
e dx e C
1
ax b ax b
e dx e C
a
ln
x
x
a
a dx C
a
22
1
ln
2
dx x a
C
a x a
xa
. Khi thay
x

()ax b

1
a
PHN 1:  HÀM
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
2 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341

1. 
2.  

3. 


 
ⓣⓗ
BÀI TP VN DNG
a)
2
( ) 3
2
a
f a a
:
2
3
( ) .
4
a
F a a C

b)
3
( ) 2 5 7.f b b b
:
42
5
( ) 7 .
22
bb
F b b C
1.  
2.  
3.  
4.  

5. Bc chn ca sin và cosin H bc.

Tìm nguyên hàm ca các hàm s
: Dựa o bng nguyên hàm của c hàm số vận dụng các tính chất
nguyên hàm.
i 1
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 3 | THBTN

c)
5 3 2
( ) 6 12 8.f c c c c
:
3
64
( ) 3 8 .
3
c
F x c c c C

d)
2
( ) ( 3 ) ( 1)f x x x x
:
4 3 2
23
( ) .
4 3 2
x x x
F x C

e)
3
( ) (3 ) .f x x
:
4
(3 )
( ) .
4
x
F x C

f)
2
2
11
()
3
f x x
x
:
3
1
( ) .
33
xx
F x C
x

g)
2
( ) 10 .
x
fx
:
2
10
( ) .
2 ln10
x
F x C

h)
3
3
( ) 4f x x x
x
:
4
2
( ) 2 3.ln .
4
x
F x x x C

i)
4
22
()
t
ft
t
:
3
22
( ) .
3
F t t C
t

j)
2
1
()
x
fx
x
:
1
( ) ln .F x x C
x

k)
2
( ) 2sin
2
x
fx
:
( ) sin .F x x x C

l)
2
( ) cos .f x x
:
11
( ) sin2 .
24
F x x x C

m)
2
( ) tan .f x x
:
( ) tan .F x x x C

n)
22
1
()
sin .cos
fx
xx
:
( ) 2cot2 .F x x C

TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
4 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
o)
( ) 2sin3 cos2 .f x x x
:
1
( ) cos 5 cos .
5
F x x x C

p)
( ) .( 1).
xx
f x e e
:
2
1
( ) .
2
xx
F x e e C

q)
2
( ) 2
cos
x
x
e
f x e
x
:
( ) 2 tan .
x
F x e x C

r)
3
( ) .I x x dx
:
2
3
3
.
2
I x C

s)
2
3
2
1
2I x dx
x
:
3
3
2
3.
3
I x x C

t)
35
1 3 5
2
I dx
xxx
:
35
24
9 25
( ) .
24
F x x x x C

u)
2
4 sin .I x dx
:
2 sin2 .I x x C

v)
1 cos 4
.
2
x
I dx
:
sin 4
.
28
xx
IC

w)
1
(3cos 3 )
x
I x dx
:
1
3
3 sin .
ln 3
x
I x C

x)
2
(tan 2 cot ) . .I x x dx
:
tan 4 cot 9 .I x x x C

y)
3
.( 4). .I u u du
:
33
74
3
3.
7
I u u C




TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 5 | THBTN
a)
32
( ) 5 4 7 120F x x x x
2
( ) 15 8 7.f x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
2
( ) ln( 3)F x x x
2
1
( ) .
3
fx
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
( ) (4 5)
x
F x x e
( ) (4 1) .
x
f x x e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
4
( ) tan 3 5F x x x
53
( ) 4 tan 4 tan 3.f x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
2
2
4
( ) ln
3
x
Fx
x
22
2
( ) .
( 4) ( 3)
x
fx
xx
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
2
2
21
( ) ln
21
xx
Fx
xx
2
4
2 2( 1)
( ) .
1
x
fx
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
Chng minh là mt nguyên hàm ca hàm s ng hp sau:
: Đ mt nguyên hàm của hàm s ta cần chứng minh:
i 2
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
6 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
a)
3
( ) 4 5, (1) 3.f x x x F
:
4
2
5
( ) 5
44
x
F x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
( ) 3 5cos , ( ) 2.f x x F
:
( ) 3 5sin 2 3 .F x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
2
35
( ) , ( ) 1.
x
f x F e
x
:
22
55
( ) 3ln 2.
22
xe
F x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
2
13
( ) , (1)
2
x
f x F
x
:
2
( ) ln 1.
2
x
F x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
1
( ) , (1) 2.f x x x F
x
:
5
2 22
( ) 2
55
F x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
sin2 .cos . ,I x x dx

0.
3
F
:
1 1 1
( ) cos 3 cos
6 2 12
F x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
Tìm nguyên hàm ca các hàm s thu king hp
sau:
 : Tìm ngun hàm của hàm s tức đi tính
Rồi sau đó thế đ tìm hng số
i 3
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 7 | THBTN
g)
43
2
3 2 5
,
xx
I dx
x

(1) 2.F
:
32
5
( ) 7.F x x x
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
32
2
3 3 7
,
( 1)
x x x
I dx
x

(0) 8.F
:
2
8
()
21
x
F x x
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
2
sin ,
2
x
I dx

24
F
:
sin 1
()
2 2 2
xx
Fx
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
j)
1
,I x x dx
x

7
(1)
2
F
:
3
13
( ) .
36
x
F x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
k)
2
2
2 cos 1
,
cos
x
I dx
x

42
F
:
( ) 2 tan 1.F x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
a)
32
2
( ) (3 2) 4 3
( ) 3 10 4
F x mx m x x
f x x x
:
1.m
.............................................................................................................................................................
u kin ca tham s m hoc a, b, c mt ngun hàm ca hàm s
: Đ mt nguyên hàm của m số T
đó, ta sử dụng đồng nhất thức đ tìm ra tham số cn tìm.
i 3
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
8 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
2
2
( ) ln 5
23
()
35
F x x mx
x
fx
xx
:
3.m
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
2
( ) ( )
( ) ( 3)
x
x
F x ax bx c e
f x x e
:
0, 1, 4.a b c
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
22
22
( ) ( )
( ) (2 8 7)
x
x
F x ax bx c e
f x x x e
:
1, 3, 2.a b c
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
2
2
( ) ( )
( ) ( 3 2)
x
x
F x ax bx c e
f x x x e
:
1, 1, 1.a b c
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
( ) ( 1)sin sin2 sin 3
23
( ) cos
bc
F x a x x x
f x x
:
0.a b c
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 9 | THBTN
g)
2
2
( ) ( ) 2 3
20 30 7
()
23
F x ax bx c x
xx
fx
x
:
4, 2, 1.a b c
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
2
( ) 3 , ( 3)
( ) ( ) 3
f x x x x
F x ax bx c x
:
2 2 12
; ;
5 5 5
a b c
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BÀI TP TRC NGHIM
NHÓM 1 : DÙNG BNG NGUYÊN HÀM
Câu 1. 
3
32f x x x

A.
42
3
2
42
xx
F x x C
. B.
4
2
32
3
x
F x x x C
.
C.
42
2
42
xx
F x x C
. D.
2
33F x x x C
.
Câu 2. 
32
5 4 7 120F x x x x C

A.
2
15 8 7f x x x
. B.
2
5 4 7f x x x
.
C.
2 3 2
5 4 7
4 3 2
x x x
fx
. D.
2
5 4 7f x x x
.
Câu 3. 
2
1
3y x x
x
A.
3
2
3
ln
32
x
F x x x C
. B.
3
2
3
ln
32
x
F x x x C
.
C.
3
2
3
ln
32
x
F x x x C
. D.
2
1
23F x x C
x
.
Câu 4. 
12f x x x
A.
3
2
3
2
32
x
F x x x C
. B.
3
2
2
2
33
x
F x x x C
.
C.
23F x x C
. D.
3
2
2
2
33
x
F x x x C
.
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
10 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
Câu 5. Nguyên hàm
Fx

2
2 2 3
52
fx
xx
x

A.
3
ln 5 2 2lnF x x x C
x
. B.
3
ln 5 2 2lnF x x x C
x
.
C.
3
ln 5 2 2lnF x x x C
x
. D.
3
ln 5 2 2 lnF x x x C
x
.
Câu 6. 
32
( ) 3 5f x x x
A.
2
36xx
B.
2
36x x C
C.
4
3
5
4
x
x x C
D.
43
5x x x C
Câu 7. 
4
( ) 3f x x
A.
4
3
4
x
B.
3
43x
C.
5
3
5
x
D.
3
3
3
x
Câu 8. 
32
( ) 3 5f x x x
A.
2
36xx
B.
2
36x x C
C.
4
3
5
4
x
x x C
D.
43
5x x x C
Câu 9. 
42
( ) 5 4 6g x x x
:
A.
53
4
6
3
x x x C
B.
3
20 8x x C
C.
3
20 8xx
D.
53
4
3
x x C
Câu 10. Tính
2
1
32x dx
x
A.
3
ln 2
3
x
x x C
B.
3
2
1
2x x C
x
C.
3
lnx x C
D.
3
ln 2x x x C
Câu 11. Cho
32
( ) 3 2f x x x x

()Fx
()fx

10F
:
A.
4
32
1
44
x
xx
B.
4
32
1
44
x
xx
C.
4
32
1
4
x
xx
D.
4
32
1
4
x
xx
Câu 12. 
()Fx

2
( ) 3 2 1f x x x
thì
()Fx
là:
A.
32
( ) 3 2F x x x x C
B.
32
( ) 1F x x x C
C.
2
( ) ( 1)F x x x x C
D.
32
11
()
32
F x x x x C
Câu 13. 
2
2
1x x dx

A.
3
2
1
()
3
x
F x C
B.
3
2
1
()
6
x
F x C
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 11 | THBTN
C.
23
()
23
xx
F x x C
D.
2
3
2
( ) 1
6
x
F x x C
Câu 14. 
Fx

1
()fx
x
:
A.
2
1
()Fx
x
B.
2
1
()Fx
x
C.
( ) lnF x x
D.
1
()Fx
x
Câu 15. 
Fx

2
( ) 3 sinf x x
x

A.
( ) 3cos 2 lnF x x x C
B.
( ) 3cos 2lnF x x x C
C.
( ) 3cos 2lnF x x x C
D.
( ) 3cos 2 lnF x x x C
Câu 16. 
Fx

2
3 3
x
f x x

A.
3
3
()
ln 3
x
F x x C
B.
3
3
()
ln 3
x
F x x C
C.
3
3
()
3 ln 3
x
x
F x C
D.
3
3
()
3 ln 3
x
x
F x C
Câu 17. Tính
9
1 x dx

A.
10
1
( ) 1
10
F x x C
B.
10
1
( ) 1
10
F x x C
C.
10
1
( ) 1
10
F x x C
D.
10
1
( ) 1
10
F x x C
Câu 18. Tìm nguyên hàm
Fx

2
2f x x x

A.
4
32
4
( ) 2
43
x
F x x x C
B.
4
2
( ) 2
4
x
F x x C
C.
3
2
( ) 2
3
x
F x x C
D. Kt qu khác.
Câu 19. 
2
( ) 2 1f x x x
:
A.
3
1
( ) 2
3
F x x x C
B.
( ) 2 2F x x C
C.
32
1
()
3
F x x x x C
D.
32
1
( ) 2
3
F x x x x C
Câu 20.  
3
( ) (2 1)f x x
là:
A.
4
1
(2 1)
2
xC
B.
4
(2 1)xC
C.
4
2(2 1)xC
D. Kt qu khác
Câu 21. 
5
( ) (1 2 )f x x
là:
A.
6
1
(1 2 )
12
xC
B.
6
(1 2 )xC
C.
6
5(1 2 )xC
D.
4
5(1 2 )xC
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
12 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
Câu 22. 
2
3
2x
x
:
A.
2
3
xC
x
B.
2
2
3
xC
x
C.
22
3lnx x C
D. Kt qu khác
Câu 23. 
fx

2 1f x x
và
15f
A.
2
3xx
B.
2
3xx
C.
2
xx
D. Kt qu khác
Câu 24. 
()y f x

2
( ) ( )( 1)f x x x x
và
(0) 3f
A.
42
( ) 3
42
xx
y f x
B.
42
( ) 3
42
xx
y f x
C.
42
( ) 3
42
xx
y f x
D.
2
( ) 3 1y f x x
Câu 25. Cho
2
( ) 3 2 3f x x x

1x
 

A.
32
( ) 3F x x x x
B.
32
( ) 3 1F x x x x
C.
32
( ) 3 2F x x x x
D.
32
( ) 3 1F x x x x
Câu 26. 
fx

2
'( ) , '(1) 0, (1) 4, ( 1) 2
b
f x ax f f f
x
A.
2
15
22
x
x
B.
2
15
22
x
x
C.
2
15
22
x
x
D. Kt qu khác
Câu 27. 
2
3
2f x x x
x
A.
3
3
4
3 ln
33
x
F x x x
C.
3
3
3
3 ln
34
x
F x x x
B.
3
3
34
33
x
F x x
x
D.
3
3
4
3 ln
33
x
F x x x
Câu 28. Nguyên hàm
Fx
32
32f x x x x

10F
:
A.
4
32
1
44
x
F x x x
B.
4
32
1
44
x
F x x x
C.
4
32
1
4
x
F x x x
D.
4
32
1
4
x
F x x x
NHÓM 2: HÀM S VÔ T ( CH
Câu 29. 
1
()
2x 1
fx
A.
x 2x 1f x d C
. B.
x 2 2x 1f x d C
.
C.
2x 1
x
2
f x d C
. D.
x 2 2x 1f x d C
.
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Câu 30. 
1
()
3
fx
x
.
A.
x 2 3f x d x C
. B.
x3f x d x C
.
C.
x 2 3f x d x C
. D.
x 3 3f x d x C
.
Câu 31. 
( ) 2x 1fx
.
A.
1
x 2x 1 2x 1
3
f x d C
. B.
2
x 2x 1 2x 1
3
f x d C
.
C.
1
x 2x 1
3
f x d C
. D.
1
x 2x 1
2
f x d C
.
Câu 32. 
( ) 5 3xfx
.
A.
2
x 5 3x 5 3x
9
f x d C
. B.
2
x 5 3x 5 3x
3
f x d
.
C.
2
x 5 3x 5 3x
9
f x d
. D.
2
x 5 3x
3
f x d C
.
Câu 33. 
3
( ) 2f x x
.
A.
3
3
x 2 2
4
f x d x x C
. B.
3
3
x 2 2
4
f x d x x C
.
C.
2
x 2 2
3
f x d x x
. D.
2
3
1
x2
3
f x d x C
.
Câu 34. 
3
( ) 1 3xfx
.
A.
3
1
x 1 3x 1 3x
4
f x d C
. B.
3
3
x 1 3x 1 3x
4
f x d C
.
C.
3
1
x 1 3x 1 3x
4
f x d C
. D.
2
3
x 1 3xf x d C
.
Câu 35. 
2
1 1 2016F x x x

A.
5
11
2
f x x x
B.
5
11
2
f x x x C
C.
2
11
5
f x x x
D.
11f x x x C
Câu 36.      
1
1
13
fx
x
 
Fx
 
2
1
3
F

Fx

A.
2
1 3x 3
3
F x x
B.
2
1 3x 3
3
F x x
C.
2
1 3x 1
3
F x x
D.
2
4 1 3x
3
Fx
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Câu 37. 
( ) 6 1F x x

()
1
a
fx
x

a

A.
3
. B.
3
. C.
6
. D.
1
6
.
Câu 38. Tính
11
2
dx
x
A.
22
xx
C
B.
2
2
x
xC
C.
11
2
2
xC
x
D.
2
2
x
C
x
Câu 39. 
1
1fx
x

A.
2
1
f x dx C
x
B.
lnf x dx x x C
C.
2
1
f x dx x C
x
D.
2
11
1
2
f x dx
x
Câu 40. 
()Fx

1
()
12
fx
x
thì
()Fx
là:
A.
1
( ) ln 1 2
2
F x x C
B.
1
( ) ln 1 2
2
F x x C
C.
( ) ln 1 2F x x C
D.
2
()
x
F x C
xx
Câu 41. 
( ) 3 4f x x

(0) 2F

A.
3
22
( ) 3 4
99
F x x
B.
3
22
( ) 3 4
99
F x x
C.
3
2 10
( ) 3 4
33
F x x
D.
3
2 10
( ) 3 4
33
F x x
Câu 42. Tìm nguyên hàm
3
2
4
x dx
x
A.
3
5
5
4 ln
3
x x C
B.
3
5
3
4 ln
5
x x C
C.
3
5
3
4 ln
5
x x C
D.
3
5
3
4 ln
5
x x C
Câu 43. 
2
()f x x k

0?k
A.
22
( ) ln
22
xk
f x x k x x k
B.
22
1
( ) ln
22
x
f x x k x x k
C.
2
( ) ln
2
k
f x x x k
D.
2
1
()fx
xk
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Câu 44. 
(I)
2
( ) 1f x x
(II)
2
( ) 1 5f x x
(III)
2
1
()
1
fx
x
(IV)
2
1
( ) - 2
1
fx
x
m s nào có mt nguyên hàm là hàm s
2
( ) ln 1F x x x
A. Ch (I) B. Ch (III) C. Ch (II) D. Ch (III) và (IV)
Câu 45. 
2
3
1
()f x x
x

A.
36
25
3 12
( ) ln
55
F x x x x x
B.
3
3
11
()
3
F x x
x
C.
2
3
()F x x x x
D.
35
26
3 12
( ) ln
55
F x x x x x
Câu 46. Nguyên hàm
2017x
x x e dx
=
A.
2017
2
5
2 2017
x
e
x x C
B.
2017
3
2
5 2017
x
e
x x C
C.
2017
2
3
5 2017
x
e
x x C
D.
2017
2
2
5 2017
x
e
x x C
NHÓM 3: HÀM S NG GIÁC
Câu 47. 
( ) cos 3
6
f x x
.
A.
1
( ) sin 3
36
f x dx x C
. B.
( ). sin 3
6
f x dx x C
.
C.
1
( ) sin 3
36
f x dx x C
. D.
1
( ) sin 3
66
f x dx x C
.
Câu 48. 
2
1a)
2
( tnfx
x
.
A.
( ) 2 tan
2
x
f x dx C
. B.
( ) tan
2
x
f x dx C
.
C.
1
( ) tan
22
x
f x dx C
. D.
( ) 2 tan
2
x
f x dx C
.
Câu 49. 
2
1
()
sin
3
fx
x
.
A.
( ) cot
3
f x dx x C
. B.
1
( ) cot
33
f x dx x C
.
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C.
( ) cot
3
f x dx x C
. D.
1
( ) cot
33
f x dx x C
.
Câu 50. Tính
sin cosx x dx
A.
cos sinx x C
B.
cos sinx x C
C.
cos sinx x C
D.
cos sinx x C
Câu 51. 
2
2
()
cos
fx
x
:
A.
2tanxC
B.
2cotxC
C.
2sinxC
D.
2cosxC
Câu 52. 
2
1
( ) 3
sin
fx
x
:
A.
3 tanx x C
B.
3 tanx x C
C.
3 cotx x C
D.
3 cotx x C
Câu 53. Cho
( ) sin cosf x x x

()Fx
()fx

0
4
F
:
A.
cos sin 2xx
B.
2
cos sin
2
xx
C.
cos sin 2xx
D.
2
cos sin
2
xx
Câu 54. 
( ) 2 sin 2cosf x x x x

()Fx
()fx

(0) 1F
là:
A.
2
cos 2sin 2x x x
B.
2
cos 2sin 2x x x
C.
2 cos 2sinxx
D.
2
cos 2sin 2x x x
Câu 55. 
2
( ) tanf x x
:
A.
3
tan
3
x
B.
3
2
tan 1
.
3
cos
x
x
C.
tanxx
D.
3
2 sin
cos
x
x
Câu 56. 
44
( ) cos sinf x x x
:
A.
cos2x
B.
1
sin 2
2
x
C.
2sin2x
D.
2
cos x
Câu 57. 
44
( ) sin cosf x x x
?
A.
1
sin 4
4
xx
B.
1
sin 4
4
xx
C.
31
sin 4
4 16
xx
D.
31
cos4
44
xx
Câu 58. 
2
sin2 3f x x x
:
A.
cos2 6F x x x
B.
1
cos2 6
2
F x x x
C.
3
1
cos2
2
F x x x
D.
3
1
cos2
2
F x x x
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Câu 59. 
2sin2f x x
?
A.
2
sinF x x
B.
2cos2F x x
C.
1
cos2
2
F x x
D.
cos2F x x
Câu 60. 
( ) sinf x x

A.
( ) cosF x x C
B.
( ) sinF x x C
C.
( ) cos 1F x x
D.
( ) sinF x x C
Câu 61. 
2
( ) 1 tanF x x dx

()Fx
:
A.
2
1
()
cos
F x C
x
B.
( ) tanF x x C
C.
( ) tanF x x C
D.
( ) cotF x x C
Câu 62. 
()Fx
 
( ) sin2f x x
thì
()Fx
là:
A.
1
( ) cos2
2
F x x C
B.
( ) 2sin2F x x C
C.
1
( ) cos2
2
F x x C
D.
( ) 2sin2F x x C
Câu 63. 
1
()Fx

2
1
( ) sinf x x

1
(0) 0F
và
2
()Fx
là nguyên

2
2
( ) cosf x x

2
(0) 0F
 
12
( ) ( )F x F x
có

A.
,
2
x k k Z
B.
,
2
x k k Z
C.
,x k k Z
D.
2,x k k Z
Câu 64. 
2
cos .siny x x
:
A.
3
1
cos
3
xC
B.
3
cos xC
C.
3
1
sin
3
xC
D. 
Câu 65. 
cos5 .cosy x x
:
A.
cos6F x x
B.
sin 6F x x
C.
1 1 1
sin 6 sin 4
2 6 4
xx
D.
1 sin 6 sin 4
2 6 4
xx
Câu 66. 
sin5 .cos 3y x x
:
A.
1 cos6 cos2
2 8 2
xx
B.
1 cos 8 cos2
2 8 2
xx
C.
cos8 cos2xx
D. 
Câu 67. 
Fx

cosf x x

A.
sinF x x C
B.
sin F x x C
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C.
cosF x x C
D.
cosF x x C
Câu 68.  
cosf x x


2
x
?
A.
sin F x x
B.
sin F x x
C.
sin 1 F x x
D.
sin 1F x x
Câu 69. Tính
2
1
cos 3 1
dx
x

A.
tan 3 1F x x C
B.
cot 3 1F x x C
C.
1
tan 3 1
3
F x x C
D.
1
cot 3 1
3
F x x C
Câu 70. 
2
tanf x x

1
4
F

A.
( ) tan
4
F x x x
B.
( ) tan
4
F x x x
C.
( ) tan
4
F x x x
D.
( ) tan
4
F x x x
Câu 71. Tính
sin(3 1)x dx

A.
1
cos(3 1)
3
xC
B.
1
cos(3 1)
3
xC
C.
cos(3 1)xC
D. Kt qu khác
Câu 72. Tìm
(cos6 cos 4 )x x dx
:
A.
11
sin6 sin 4
64
x x C
B.
6sin6 5sin4x x C
C.
11
sin6 sin 4
64
x x C
D.
6sin6 sin4x x C
Câu 73. 
(I)
2
( ) tan 2f x x
(II)
2
2
()
cos
fx
x
(III)
2
( ) tan 1f x x
m s nào có mt nguyên hàm là hàm s g(x) = tanx
A. (I), (II), (III) B. Ch (II), (III) C. Ch (III) D. Ch (II)
Câu 74. 
2sin 3 cos2f x x x
A.
1
cos5 cos
5
x x C
B.
1
cos 5 cos
5
x x C
C.
5cos5 cosx x C
D. Kt qu khác
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Câu 75. 
A.
cot ln sinxdx x C
B.
sin cosxdx x C
C.
2
11
dx C
x
x
D.
cos sinxdx x C
Câu 76. Tìm nguyên hàm
2
(1 sin )x dx
A.
21
2 cos sin2
34
x x x C
B.
21
2 cos sin2
34
x x x C
C.
21
2 cos2 sin2
34
x x x C
D.
31
2 cos sin2
24
x x x C
Câu 77. Trong 
23
2
2
11
( ) sin sin 3 (sin2 - sin 4 )
42
1
( ) tan tan
3
11
( ) ln( 2 3)
2
23
I x xdx x x C
II xdx x C
x
III dx x x C
xx
A. Ch (I) và (II) B. Ch (III) C. Ch (II) và (III) D. Ch (II)
Câu 78. Tìm
3
(sin 1) cosx xdx
:
A.
4
(cos 1)
4
x
C
B.
4
sin
4
x
C
C.
4
(sin 1)
4
x
C
D.
3
4(sin 1)xC
Câu 79. 
(I)
( ) cosF x x x
mt nguyên hàm ca
2
( ) sin - cos
22
xx
fx
(II)
4
( ) 6
4
x
F x x
mt nguyên hàm ca
3
3
()f x x
x
(III)
( ) tanF x x
mt ngun hàm ca
( ) - ln cosf x x
M nào sai ?
A. (I) và (II) B. Ch (III) C. Ch (II) D. Ch (I) và (III)
Câu 80. Tìm
3
sin
2
x
dx
.
A.
23
( ) cos
32
x
F x C
B.
23
( ) cos
32
x
F x C
C.
33
( ) cos
22
x
F x C
D.
33
( ) cos
22
x
F x C
Câu 81. 
3
sin .cosy x x
là:
A.
4
1
( ) sin
4
F x x C
B.
4
1
( ) sin
4
F x x C
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C.
4
1
( ) cos
4
F x x C
D.
4
1
( ) cos
4
F x x C
Câu 82. Nguyên hàm
Fx

2
tanf x x

03F
:
A.
tan 3f x x x
B.
tan 3f x x x
C.
tan 3f x x x
D.
tan 3f x x x
Câu 83. 
2
cos
4
f x x
và
13
0
4
f
thì:
A.
11
cos2 3
22
f x x x
B.
1
cos2 4
2
f x x x
C.
7
sin
2
f x x
D.
1
cos2 3
2
f x x
Câu 84. 
sin cos 1y f x x x
:
A.
sin cosF x x x C
B.
sin cosF x x x x C
C.
cos sinF x x x x C
D.
sin cosF x x x x C
Câu 85. 
A.
sin .cos cos .sinx xdx x x C
B.
1
sin .cos cos2
2
x xdx x C
C.
3
2
cos
cos .sin
3
x
x xdx C
D.
3
2
sin
sin .cos
3
x
x xdx C
Câu 86. 
A.
1 1 1
cos 3 .cos sin 4 sin2
2 4 2
x xdx x x C
B.
1 1 1
sin 3 .cos cos 4 cos2
2 4 2
x xdx x x C
C.
1
sin 3 .cos cos 3 .sin
3
x xdx x x C
D.
cos2
sin .cos
4
x
x xdx C
Câu 87. y =
22
cos2
sin .cos
x
dx
xx
:
A.
cos sin F x x x C
B.
cos sin F x x x C
C.
cot tan F x x x C
D.
cot tan F x x x C
Câu 88. Tìm nguyên hàm
2sin 3 cos2 .xxdx
?
A.
1
cos5 cos
5
xxFx C
B.
11
cos5 cos
32
F x xx C
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C.
11
cos5 cos
23
xCF xx
D.
1
cos5 cos
5
x x CFx
Câu 89. Tìm nguyên hàm:
2
sin 2xdx
A.
11
sin 4
28
x x C
B.
3
1
sin 2
3
xC
C.
11
sin 4
28
x x C
D.
11
sin 4
24
x x C
Câu 90. Tìm nguyên hàm
22
1
sin .cos
dx
xx
=
A.
2tan2xC
B. 2
cot2xC
C. 4
cot2xC
D. 2
cot2xC
Câu 91. 
A.
2
1
cos 2 sin 4
28
x
xdx x C
B.
2
1
sin 2 sin 4
28
x
xdx x C
C.
1
cos 4 sin 4
4
xdx x C
D.
22
sin 2 cos 2xdx x C
Câu 92. 
A.
2
cot cotxdx x x C
B.
2
tan tanxdx x x C
C.
3
2
tan
tan
3
x
xdx C
D.
2
1 sin2
sin
22
x
xdx x C
Câu 93. 
fx

' sin cosf x x x
và
0
4
f
.
A.
cos sin 2f x x x
` B.
cos sin 2f x x x
C.
2
cos sin
2
f x x x
D.
2
cos sin
2
f x x x
NHÓM 4: HÀM S 
Câu 94. 
()
xx
f x e e
.
A.
x
xx
f x d e e C
. B.
x
xx
f x d e e C
.
C.
x
xx
f x d e e C
. D.
x
xx
f x d e e C
.
Câu 95. 
2x
( ) 2 .3
x
fx
.
A.
21
x.
9 ln 2 ln 9
x
f x d C
. B.
91
x.
2 ln2 ln 9
x
f x d C
.
C.
21
x.
3 ln 2 ln 9
x
f x d C
. D.
21
x.
9 ln 2 ln 9
x
f x d C
.
Câu 96. 
( ) (3 )
xx
f x e e
A.
( ) 3
x
F x e x C
. B.
( ) 3 ln
x x x
F x e e e C
.
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C.
1
( ) 3
x
x
F x e C
e
. D.
( ) 3
x
F x e x C
.
Câu 97. 
7 tan
x
F x e x

A.
2
7
cos
x
x
e
f x e
x
. B.
2
1
7
cos
x
f x e
x
.
C.
2
7 tan 1
x
f x e x
. D.
2
1
7
cos
x
f x e
x
.
Câu 98. 
4x 2
()f x e
.
A.
2x 1
1
x
2
f x d e C
. B.
2x 1
xf x d e C
.
C.
4x 2
1
x
2
f x d e C
. D.
2x 1
1
x
2
f x d e C
.
Câu 99. Tính
4
x
e dx
A.
4
x
e x C
B.
1
4
x
xC
e
C.
x
eC
D.
4
x
e x C
Câu 100. Tính
31
2
1
x
e dx
x
A.
31
11
3
x
eC
x
B.
31
1
3
x
eC
x
C.
31
1
3
x
eC
x
D.
31
11
3
x
eC
x
Câu 101. 
2
93
x
f x x
A.
3
9
x
F x x
B.
3
9 ln 9
x
F x x
C.
3
9
9
x
F x x
D.
3
9
ln 9
x
F x x
Câu 102. 
Fx

2
1
x
x
e
f x e
x

1Fe
A.
1
1
x
F x e
x
B.
1
1
x
F x e
x
C.
1
1
x
F x e
x
D.
1
1
x
F x e
x
Câu 103. 
1
()
x
f x e

A.
1
()
x
F x e C
B.
1
()
x
F x e
C.
2
1
()
2
x
F x e C
x
D.
1
()
x
F x e C
Câu 104. 
()Fx

1
( ) 3
x
fx
thì
()Fx
là:
A.
1
( ) 3
x
F x C
B.
( ) 3ln 3.3
x
F x C
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C.
1
( ) 3ln 3.3
x
F x C
D.
1
3
()
ln 3
x
F x C
Câu 105. 
()Fx

( ) 2
x
f x e
thì
()Fx
là:
A.
( ) 2
x
F x e C
B.
( ) 2
x
F x e x C
C.
1
( ) 2
x
F x e x C
D.
1
1
( ) 2
1
x
F x e x C
x
Câu 106. 
2
()
xx
f x e e
là:
A.
2
1
2
xx
e e C
B.
2
2
xx
e e C
C.
()
xx
e e x C
D. Kt qu khác
Câu 107. 
2
1
( ) 2
cos
x
f x e
x
là:
A.
2 tan
x
e x C
B.
2
2
cos
x
x
e
x
ex
C.
tan
x
e x C
D. Kt qu khác
Câu 108. Tính
(3cos 3 )
x
x dx

A.
3
3 sin
ln 3
x
xC
B.
3
3 sin
ln 3
x
xC
C.
3
3 sin
ln 3
x
xC
D.
3
3 sin
ln 3
x
xC
Câu 109. 
tan
x
F x e x C

()fx
nào?
A.
2
1
()
sin
x
f x e
x
B.
2
1
()
sin
x
f x e
x
C.
2
1
()
cos
x
f x e
x
D. Kt qu khác
Câu 110. 
( ) sin2
x
f x dx e x C
thì
()fx

A.
cos2
x
ex
B.
cos2
x
ex
C.
2cos2
x
ex
D.
1
cos2
2
x
ex
Câu 111. 
24
xx
fx
.
A.
2
2
2
()
ln2 ln 2
x
x
F x C
B.
1
2
( ) 1 2
ln2
x
x
F x C
C.
24
( ) 1
ln2 ln 2
xx
F x C
D.
2
( ) 1 2
2 ln2
x
x
F x C
.
Câu 112. Tìm
31
2
1
x
e dx
x
.
A.
31
11
()
3
x
F x e C
x
B.
31
1
( ) 3
x
F x e C
x
C.
31
1
( ) 3
x
F x e C
x
D.
31
11
()
3
x
F x e C
x
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Câu 113. Tìm
5
2
3
x
x
e
xe
.
A.
4
1
( ) 3
2
x
F x e C
x
B.
4
1
( ) 3
2
x
F x e C
x
C.
4
1
( ) 3
2
x
F x e C
x
D.
4
1
( ) 3
2
x
F x e C
x
Câu 114. 
23
21
x
y f x e x
:
A.
2 3 2x
F x e x x C
B.
23
2
2
x
e
F x x C
C.
23
2
2
x
e
F x x x C
D.
2 3 2
2
x
F x e x x C
Câu 115. 
3 2 1
24
xx
y f x
A
3 2 1
24
2.ln 8 4.ln16
xx
F x C
B.
3 2 1
24
ln2 ln 4
xx
F x C
C.
3 2 1
24
ln 8 ln16
xx
F x C
D.
3 2 1
3.2 2.4
ln2 ln 4
xx
F x C
Câu 116. 
A.
22
x x x x
e dx e C
B.
2
2
ln2
x
x x x
e dx e C
C.
3
2
1
1
3
x
xx
e
e e dx C
D.
11
xx
xdx C
ee
Câu 117. Tìm nguyên hàm
2
3
2
x
e dx
A.
36
4
3
36
xx
ee
xC
B.
36
4
4
36
xx
ee
xC
C.
36
45
4
36
xx
ee
xC
D.
36
4
4
36
xx
ee
xC
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 25 | THBTN
 
ⓣⓗ
BÀI TP VN DNG
BT 1. 
a)
21
1
x
I dx
x
:
2 3 ln 1 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
b)
31
2
x
I dx
x
:
3 7 ln 2 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................

  

 :
  
  







TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
26 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
c)
3
31
( 1)
x
I dx
x
:
2
31
.
1
( 1)
IC
x
x
.............................................................................................................................................................
.............................................................................................................................................................
d)
1
23
x
I dx
x
:
1
ln 2 3 .
24
x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
e)
42
2
3 2 1x x x
I dx
x
:
3
1
3 2 ln .
3
x
I x x C
x
.............................................................................................................................................................
.............................................................................................................................................................
f)
2
1
2
xx
I dx
x
:
2
3 ln 2 .
2
x
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
g)
2
4 6 1
21
xx
I dx
x
:
2
1
2 ln 2 1 .
2
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
h)
32
4 4 1
21
xx
I dx
x
:
32
21
ln 2 1 .
3 2 2 2
x x x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
i)
( 1)
dx
I
xx
:
ln .
1
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
j)
2
4
dx
I
x
:
12
ln .
42
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
k)
2
69
dx
I
xx
:
1
.
3
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 27 | THBTN
l)
2
65
dx
I
xx
:
15
ln .
41
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
m)
2
23
dx
I
xx
:
11
ln .
5 2 3
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
n)
2
2 7 5
dx
I
xx
:
1 2 5
ln .
31
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
o)
2
45
2
x
I dx
xx
:
ln 2 3ln 1 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
p)
2
4 11
56
x
I dx
xx
:
3ln 2 ln 3 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
q)
2
1
6
x
I dx
xx
:
1
ln 2 4 ln 3 .
5
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
r)
2
53
32
x
I dx
xx
:
2ln 1 7 ln 2 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
s)
2
12
2
x
I dx
xx
:
13
ln ln 2 .
22
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
t)
2
2
7 12
x dx
I
xx
:
16ln 4 9 ln 3 .I x x x C
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
28 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
.............................................................................................................................................................
u)
2
2
1
1
x
I dx
x
:
1
ln .
1
x
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
v)
2
2
x
I dx
x
:
2
1
2 4 ln 2 .
2
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
w)
2
32
4 4 1
x
I dx
xx
:
37
ln 2 1 .
4 4(2 1)
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
x)
24
2
3
2
xx
I dx
x
:
3
22
ln .
32
2
xx
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
y)
2
2
( 2)
xx
I dx
x
:
2
3 ln 2 .
2
I x x C
x
.............................................................................................................................................................
.............................................................................................................................................................
z)
2
22
.
(1 )
x dx
I
x
:
1 1 1 1
ln .
4 1 1 1
x
IC
x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 2. 
a)
2
32
2 5 3
2
xx
I dx
x x x
:
35
ln 2 ln 1 ln 2 .
22
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 29 | THBTN
b)
2
32
2 8 10
44
xx
I dx
x x x
:
1 20 17
ln 2 ln 1 ln 2 .
6 3 2
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
3
32
1
56
x
I dx
x x x
:
1 9 28
ln ln 2 ln 3 .
6 2 3
I x x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
2
3
3 3 3
32
xx
I dx
xx
:
3
2ln 1 ln 2 .
1
I x x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
3
( 1)
dx
I
xx
:
3
1
ln ln 1 .
3
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BÀI TP TRC NGHIM
Câu 1. ên hàà
35
2
x
y
x
:
A.
( ) 3 4 ln 2F x x x C
B.
( ) 3 ln 2F x x x C
C.
( ) 3 ln 2F x x x C
D.
( ) 3 ln 2F x x x C
Câu 2. 
()
1
x
fx
x
:
A.
ln 1x
B.
ln 1xx
C.
ln 1xx
D.
2ln 1x
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
30 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
Câu 3. 
2
2
21
()
21
xx
fx
xx

()Fx
()fx

(1) 0F
:
A.
2
2
1
x
x
B.
2
2
1
x
x
C.
2
2ln 1xx
D.
2
2
1
x
x
Câu 4. 
2
2
1
xx
fx
x
?
A.
2
1
1
xx
x
B.
2
1
1
xx
x
C.
2
1
x
x
D.
2
1
1
xx
x
Câu 5. 
2
2
3
1x
fx
x

Fx
fx

14F
:
A.
2
2
2
2 ln 4
2
x
x
x
B.
2
2
1
2 ln 4
2
2
x
x
x
C.
2
2
2
2 ln 4
2
x
x
x
D.
3
2F x x x C
Câu 6. 
3
1
1
x
fx
x
:
A.
32
2 ln 1
32
xx
F x x x C
B.
32
2 ln 1
32
xx
F x x x C
C.
32
ln 1
32
xx
F x x x C
D.
32
2 ln 1
32
xx
F x x x C
Câu 7. 
()Fx

32
2
3 3 1
()
21
x x x
fx
xx

1
(1)
3
F

()Fx
:
A.
2
2 13
()
2 1 6
x
F x x
x
B.
2
2 13
()
2 1 6
x
F x x
x
C.
2
1
()
21
x
F x x C
x
D.
2
2
()
21
x
F x x
x
Câu 8. 
2
21
()
xx
fx
x

1
(1)
2
F

A.
2
( ) 2 ln 2
2
x
F x x x
B.
2
( ) 2 ln 2
2
x
F x x x
C.
2
1
( ) 2 ln
22
x
F x x x
D.
2
1
( ) 2 ln
22
x
F x x x
Câu 9. Ta có:
2
32
3
3 3 3
( ) 2
12
32
1
1
A
x x A B C
f x B
xx
xx
x
C
.
Tính
( ) ( )f x dx F x C

TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 31 | THBTN
A.
2
3 2 1
()
12
1
F x C
xx
x
B.
3
( ) 2 ln 1 ln 2
1
F x x x C
x
C.
2
( ) 3 ln 1 ln 2
1
F x x x C
x
D.
1
( ) 3ln 1 2 ln 2
1
F x x x C
x
Câu 10. 
2
11
()fx
x
x
là :
A.
2
ln lnx x C
B.
1
lnxC
x
C.
1
lnxC
x
D. Kt qu khác
Câu 11. Tính nguyên hàm
1
21
dx
x

A.
1
ln 2 1
2
xC
B.
ln 2 1xC
C.
1
ln 2 1
2
xC
D.
ln 2 1xC
Câu 12. 
fx
=
4
2
23x
x
:
A.
3
23
3
x
C
x
B.
3
2
23
3
x
C
x
C.
3
2
2
3 ln
3
x
xC
D. Kt qu khác
Câu 13. 
2
1
x
dx
x
:
A.
2
1 xC
B.
2
1
1
C
x
C.
2
1
1
C
x
D.
2
1 xC
Câu 14. 
1
x
fx
x
là:
A.
( ) ln 1F x x
B.
( ) ln 1F x x x
C.
( ) ln 1F x x x
D.
( ) 2 ln 1F x x
Câu 15. 
44
sin cosf x x x
?
A.
1
( ) sin 4
4
F x x x
B.
1
( ) sin 4
4
F x x x
C.
31
( ) sin 4
4 16
F x x x
D.
31
( ) cos 4
44
F x x x
Câu 16. F(x
1
25
fx
x
A.
1
( ) ln 2 5 2016
2
F x x
B.
F(x) ln 2 5x
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
32 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
C.
2
2
()
25
Fx
x
D.
2
1
()
25
Fx
x
Câu 17. 
2
1
12
y f x
x
:
A.
11
.
2 1 2
F x C
x
B.
2
ln 1 2F x x C
C.
11
.
2 1 2
F x C
x
D.
1
12
F x C
x
Câu 18. 
1
sin2
43
y f x x
x
:
A.
11
.cos2 .ln 4 3
24
F x x x C
B.
11
.cos2 .ln 4 3
24
F x x x C
C.
1
cos2 .ln 4 3
4
F x x x C
D.
1
.cos2 ln 4 3
2
F x x x C
Câu 19. 
A.
4
4 ln 2 1
21
dx x C
x
B.
21
2 ln
x
dx x x C
x
C.
2
ln 2 1
21
dx x C
x
D.
21
2 ln 1
1
x
dx x x C
x
Câu 20. 
A.
2 1 5
ln 3 4
3 4 2 8
xx
dx x C
x
B.
1
2ln 1
1
x
dx x x C
x
C.
2
2 ln 1
1
dx x C
x
D.
6
3 ln 2 3
23
dx x C
x
Câu 21. 
3
1
x
y dx
x
:
A.
32
11
ln 1
32
x x x x C
B.
32
11
ln 1
32
x x x x C
C.
32
11
ln 1
62
x x x x C
D.
32
11
ln 1
34
x x x x C
Câu 22. 
2
.
23
21
I
x
dx
xx
:
A. F(x) =
25
ln 2 1 ln 1
33
x x C
B. F(x) =
25
ln 2 1 ln 1
52
x x C
C. F(x) =
25
ln 2 1 ln 1
33
x x C
D. F(x) =
25
ln 2 1 ln 1
33
x x C
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 33 | THBTN
  
ⓣⓗ
A I TOÁN
: Cho
( ) ( )f u du F u C

()u u x

( ) ( ) ( ) .f u x u x dx F u x C
1. 
( ).tx
1
1
22
( ) .
1 ( 1) . ,
1
( ) 2 .
PP
n
m
n
PP
nn
n
PP
n
I f ax b xdx t ax b dt a dx
x
I dx t x dt n x dx
ax
I f ax b xdx t ax b dt ax dx

, .mn
( ) ( )
n
I f x f x dx
PP

( ),
n
t f x
     

1
(ln )
1
( ln )
I f x dx
x
I f a b x dx
x
PP

ln
ln
tx
t a b x
()
xx
I f e e dx
PP

.
x
te
(cos ) sinI f x xdx
PP

cos sin .t x dt xdx
(sin ) cosI f x xdx
PP

sin cos .t x dt xdx
2
1
(tan )
cos
I f x dx
x
PP

2
2
1
tan (1 tan ) .
cos
t x dt dx x dx
x
2
1
(cot )
sin
I f x dx
x
PP

2
2
1
cot (1 cot ) .
sin
t x dt dx x dx
x
22
(sin ;cos ) sin2I f x x xdx
PP

2
2
sin sin2
cos sin 2
t x dt xdx
t x dt xdx
(sin cos ) (sin cos )I f x x x x dx
PP

sin cos .t x x
2. 
( ).xt
2 2 2
()
n
I f a x x dx
PP

.sin .cos . .x a t dx a t dt
2 2 2
()
n
I f x a x dx
PP

2
.tan
cos
adt
x a t dx
t
2 2 2
()
n
I f x a x dx
PP

2
sin
cos
cos
a a t
x dx dt
t
t
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
34 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
2
( ) .
n
dx
I
x a ax bx c
PP

2
1 dt
x a dx
t
t
1
,...,
k
nn
I R ax b ax b dx
PP

n
t ax b
12
. . . ; ;...;
k
n BC N N n n n
( )( )
dx
I
x a x b
PP

0
khi
0
0
khi
0
xa
t x a x b
xb
xa
t x a x b
xb
B - BÀI TP VN DNG
BT 3. 
a)
2015
(1 )I x x dx
:
2016 2017
(1 ) (1 )
.
2016 2017
xx
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
23
( 1)I x x dx
:
4
2
( 1)
.
8
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
29
( 1)I x x dx
:
12 11 10
( 1) 2( 1) ( 1)
.
12 11 10
x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
3 2 8
(2 3 )I x x dx
:
2 10 2 9
(2 3 ) (2 3 )
.
180 81
xx
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 35 | THBTN
e)
2
2
xdx
I
x
:
2
1
ln 2 .
2
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
2
2
( 1)
x
I dx
x
:
2
2 ln 1 .
1
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
5
( 1)
x
I dx
x
:
3
1 1 1 1
.
4 1 3
( 1)
IC
x
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
3
2
1
x
I dx
x
:
2 2 2
11
.
2(1 ) 4(1 )
IC
xx
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
3
2
11
1I dx
x
x
:
4 3 2
1 1 3 1
.
42
IC
x
x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
j)
2
( 1)
xdx
I
x
:
1
ln 1 .
1
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
36 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
.............................................................................................................................................................
k)
3
(2 1)
xdx
I
x
:
2
1 1 1
.
2 2(2 1)
4(2 1)
IC
x
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
l)
5
2
1
x
I dx
x
:
42
2
1
ln( 1) .
4 2 2
xx
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
m)
4
10
4
x
I dx
x
:
5
5
12
ln .
20
2
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
n)
3
1
x dx
I
x
:
32
ln 1 .
32
xx
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 4. 
a)
2
( 1)
24
x dx
I
xx
:
2
2 4 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
2
. 2 . .I x x dx
:
23
(2 )
.
3
x
IC
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 37 | THBTN
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
2
3
2
4
xdx
I
x
:
22
3
3
( 4) .
2
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
2
1
x dx
I
x
:
2
2(3 4 8) 1
.
15
x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
3
2
5 . 1 . .I x x dx
:
4
2
3
15
(1 ) .
8
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
5 2 2
3
. (1 2 ) . .I x x dx
:
4 2 2 2
3
3(20 4 3). (1 2 )
.
320
x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
41
.
2 1 2
x
I dx
x
:
2 1 4 2 1 5ln 2 1 2 .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
38 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
h)
3
2
.
4
x
I dx
x
:
23
2
(4 )
4 4 .
3
x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
2
4
dx
I
xx
:
2
2
1 4 2
ln .
4
42
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
j)
32
2
23
.
1
x x x
I dx
xx
:
23
2
2 ( 1)
2 1 .
3
xx
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
k)
3
sin . cos . .I x x dx
:
3
2
(3 cos 7 cos ) cos .
21
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
l)
1
x
dx
I
e
:
11
ln .
11
x
x
e
IC
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
m)
2
ln . 1 3 ln
dx
I
x x x
:
2
2
1 1 3 ln 1
ln .
2
1 3 ln 1
x
IC
x
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 39 | THBTN
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
n)
2
1
xdx
I
xx
:
23
3
( 1)
.
33
x
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 5. 
a)
2
1
lnI x dx
x
:
3
ln
.
3
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
3 ln 1
ln
x
I dx
xx
:
3ln ln ln .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
1
(1 ln )I x dx
x
:
2
(1 ln )
.
2
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
ln 1
1 ln
x
I dx
x
x
:
3
2 (1 ln )
2 1 ln .
3
x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
40 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
e)
2
3
ln . 2 lnx xdx
I
x
:
24
3
3
(2 ln ) .
8
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
3
2
2
log
1 3 ln
x
I dx
xx
:
23
2
3
(1 3 ln )
1
1 3 ln .
3
9 ln 2
x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 6. 
a)
1
x
dx
I
e
:
1
ln .
x
x
e
IC
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
2
x
dx
I
e
:
1
ln .
2
2
x
x
e
IC
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
23
xx
dx
I
ee
:
2
ln .
1
x
x
e
IC
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
x
xx
e
I dx
ee
:
2
1
ln 1 .
2
x
I e C
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 41 | THBTN
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
4.
xx
dx
I
ee
:
12
ln .
4
2
x
x
e
IC
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
3
(1 )
x
x
e
I dx
e
:
2
(1 ) 1
2(1 ) 3 .
2
x
x
x
e
I e x C
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
2
2
3
32
xx
xx
ee
I dx
ee
:
2
1 3 1
ln( 3 2) ln .
22
2
x
xx
x
e
I e e C
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
2
1
x
x
e
I dx
e
:
3
2 ( 1)
2 1 .
3
x
x
e
I e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
xx
dx
I
ee
:
12
2 ln .
x
xx
e
IC
ee
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
42 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
j)
1
x
dx
I
e
:
2 ln .
1
x
x
e
IC
e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 7. 
a)
cos
1 sin
xdx
I
x
:
ln 1 sin .I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
(2sin 3)cos
2 sin 1
xx
I dx
x
:
1
(2sin 1) 2 ln 2 sin 1 .
2
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
2
3 cos
(1 sin )
xdx
I
x
:
3
.
1 sin
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
2 cos
3 2 sin
xdx
I
x
:
3
ln sin .
2
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
23
sin .cos . .I x x dx
:
35
sin sin
.
35
xx
IC
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
.............................................................................................................................................................
f)
2
1 2 sin
1 sin2
x
I dx
x
:
1
ln 1 sin2 .
2
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
3
2
cos
sin
x
I dx
x
:
1
sin .
sin
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
2
sin2
(2 sin )
x
I dx
x
:
4
2 ln(2 sin ) .
2 sin
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
sin
.cos . .
x
I e x dx
:
sin
.
x
I e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
j)
32
(cos 1).cos . .I x x dx
:
53
sin 2 sin sin2
sin .
5 3 2 4
x x x x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
k)
2
cos
6 5 sin sin
xdx
I
xx
:
3 sin
ln .
2 sin
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
.............................................................................................................................................................
l)
2
cos
11 7 sin cos
xdx
I
xx
:
1 5 sin
ln .
3 2 sin
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 8. 
a)
sin
1 cos
xdx
I
x
:
ln 1 cos .I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
3
4 sin
1 cos
x
I dx
x
:
2
4cos cos2 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
23
cos sinI x xdx
:
53
cos cos
.
53
xx
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
sin2 cos
1 cos
xx
I dx
x
:
2
2cos cos ln cos 1 .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
2
sin 4
1 cos
x
I dx
x
:
3 cos2
6ln( ) 2cos2 6 .
2
x
I x C
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.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
sin sin 3
cos2
xx
I dx
x
:
2 2 cos 1
ln 2 cos .
2
2 cos 1
x
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
3
tan .sin .(1 sin )
42
cos
x
xx
I dx
x
:
1
.
cos
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
3
4
sin
cos
x
I dx
x
:
3
11
.
cos
3 cos
IC
x
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 9. 
a)
2
tan
cos
x
I dx
x
:
2
tan
.
2
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
4
6
sin
cos
x
I dx
x
:
5
tan
.
5
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
c)
4
tan
cos2
x
I dx
x
:
3
tan 1 tan 1
tan ln .
3 2 tan 1
xx
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
22
5 cos 8 sin cos 3sin
dx
I
x x x x
:
1 3 tan 5
ln .
2 tan 1
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
34
(1 sin2 )
2 sin cos cos
x dx
I
x x x
:
2
tan 3tan 1
ln 2 tan 1 .
4 4 8
xx
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
42
cos sin
dx
I
xx
:
3
tan 1
2 tan .
3 tan
x
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
cos cos
4
dx
I
xx
:
2 ln 1 tan .I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
tan
4
cos2
x
I dx
x
:
1
.
1 tan
IC
x
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.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 10. 
a)
2
4
cos
sin
x
I dx
x
:
3
1
cot .
3
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
2
8
cos
sin
x
I dx
x
:
7 5 3
15cot 42 cot 35cot
.
105
x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
4
2
sin cot
dx
I
xx
:
4
3
4
cot .
3
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
3
cos sin
dx
I
xx
:
2
1
ln cot cot .
2
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
sin sin
6
dx
I
xx
:
2ln cot .I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
f)
3
sin .
(sin cos )
x dx
I
xx
:
2
1
.
2(1 cot )
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 11. 
a)
sin cos
sin cos
xx
I dx
xx
:
ln sin cos .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
cos2
sin cos 2
xdx
I
xx
:
sin cos 2 2 ln sin cos 2I x x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
3
cos2
(sin cos 2)
x dx
I
xx
:
2
11
.
sin cos 2
(sin cos 2)
IC
xx
xx
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
23
sin2 .(1 sin ) . .I x x dx
:
24
(1 sin )
.
4
x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 12. 
a)
2
1
dx
I
x
:
arcsin .I x C
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.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
22
9
dx
I
xx
:
2
9
.
9
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
23
( 1)
dx
I
x
:
2
.
1
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
22
25
dx
I
xx
:
2
25
.
25
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
3
2
1
x
I dx
x
:
22
1
( 2) 1 .
3
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
2
4
1 x
I dx
x
:
3
2
2
3
(1 )
.
3
x
IC
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
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g)
2
4
dx
I
x
:
2
ln 4 .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
C - BÀI TP TRC NGHIM
Câu 118. Tính
2
1
25
x
dx
xx
A.
2
22
25
x
C
xx
B.
2
2 2 5x x C
C.
2
25
2
xx
C
D.
2
25x x C
Câu 119. 
2
1
x
fx
x
A.
2
ln 1F x x C
B.
2
1F x x C
C.
2
21F x x C
D.
2
2
31
F x C
x
Câu 120. 
sin
cos .
x
f x x e
A.
sinx
F x e
B.
cosx
F x e
C.
sinx
F x e
D.
sin
sin .
x
F x x e
Câu 121. 
2016
2
1f x x x

A.
2017
2
1
4034
x
f x dx C
B.
2016
2
1
4032
x
f x dx
C.
2016
2
1
2016
x
f x dx
D.
2017
2
1
2017
x
f x dx
Câu 122. 
2
x
F x e

A.
2
2
x
f x xe
B.
C.
2
2
x
e
fx
x
D.
2
2
1
x
f x x e
Câu 123. 
( ) tanf x x
:
A.
2
1
()
cos
F x C
x
B.
2
1
()
sin
F x C
x
C.
( ) ln cosF x x C
D.
( ) ln cosF x x C
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Câu 124. 
cos s in 1x x dx

A.
3
2
( ) s in 1
3
F x x C
B.
3
2
( ) s in 1
3
F x x C
C.
2
( ) s in 1
3
F x x C
D.
3
2
( ) sin 1
3
F x x C
Câu 125. 
3
x
x
e
dx
e

A.
( ) 3
x
F x e C
B.
( ) 2 3
x
F x e C
C.
( ) 3
x
F x e C
D.
()
3
x
x
e
F x C
ex
Câu 126. 
ln
()
x
fx
x
có các nguyên hàm là:
A.
2
( ) lnF x x C
B.
1
( ) ln
2
F x x C
C.
2
1
( ) ln
2
F x x C
D.
2
1
()
.
F x C
xx
Câu 127. 
1
( ) ln ( )
ln
x
f x x
xx
có các nguyên hàm là:
A.
22
( ) lnF x x x C
B.
22
ln
()
2
xx
F x C
C.
2
2
ln
()
2
x
F x x C
D.
2
( ) ln (ln )
2 ln
x
F x x x C
x
Câu 128. 
()Fx

1
()
s in
fx
x

1
2
F

()Fx
:
A.
1 1 cos
( ) ln 1
2 1 cos
x
Fx
x
B.
1 1 cos
( ) ln
2 1 cos
x
Fx
x
C.
1 cos
( ) ln 1
1 cos
x
Fx
x
D.
1 1 cos
( ) ln 1
2 1 cos
x
Fx
x
Câu 129. 
()Fx
   
2
()
8
x
fx
x
 
(2) 0F
 
trình
()F x x

A.
0x
B.
1x
C.
1x
D.
13x
Câu 130. y = sin
2
x.cos
3
x là:
A.
35
11
sin sin
35
x x C
B.
35
11
sin sin
35
x x C
C. sin
3
x sin
5
x + C D. 
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Câu 131. 
3
2
2
x
y
x
:
A.
2
( ) 2F x x x
B.
22
1
42
3
xx
C.
22
1
2
3
xx
D.
22
1
42
3
xx
Câu 132. 
2
1
4
y
x
A.
2
( ) ln 4F x x x
B.
2
( ) ln 4F x x x
C.
2
( ) 2 4F x x
D.
2
( ) 2 4F x x x
Câu 133. 
2
( ) sin 1f x x x
:
A.
2 2 2
( ) 1 cos 1 sin 1F x x x x
B.
2 2 2
( ) 1 cos 1 sin 1F x x x x
C.
2 2 2
( ) 1 cos 1 sin 1F x x x x
D.
2 2 2
( ) 1 cos 1 sin 1F x x x x
Câu 134. 
2
( ) 1f x x x
:
A.
2
2
1
( ) 1
2
F x x
B.
3
2
1
( ) 1
3
F x x
C.
2
2
2
( ) 1
2
x
F x x
D.
2
2
1
( ) 1
3
F x x
Câu 135. 
2015
1f x x x

A.
2017 2016
11
( ) 1 1
2017 2016
F x x x C
B.
2017 2016
11
( ) 1 1
2017 2016
F x x x C
C.
2017 2016
11
( ) 1 1
2017 2016
F x x x C
D.
2017 2016
11
( ) 1 1
2017 2016
F x x x C
Câu 136. Tìm nguyên hàm
Fx

2
2
()
1
x
fx
xx

A.
3 2 2
22
( ) 1 1
33
F x x x x
B.
3 2 2
22
( ) 1 1
33
F x x x x
C.
3 2 2
22
( ) 1 1
33
F x x x x
D.
3 2 2
22
( ) 1 1
33
F x x x x
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Câu 137. Tính:
2
1x
P dx
x
A.
2
1P x x x C
B.
22
1 ln 1P x x x C
C.
2
2
11
1 ln
x
P x C
x
D.  án khác.
Câu 138. Tìm nguyên hàm
Fx

sin
()
sin cos
x
fx
xx

A.
1
( ) ln sin cos
2
F x x x x C
B.
1
( ) ln sin cos
2
F x x x x C
C.
1
( ) ln sin cos
2
F x x x x C
D.
1
( ) ln sin cos
2
F x x x x C
Câu 139. Tìm nguyên hàm
Fx

cos cos2 sin 4f x x x x

A.
1 1 1 1
( ) cos7 cos5 cos3 cos
28 20 12 4
F x x x x x C
B.
1 1 1 1
( ) cos 7 cos5 cos 3 cos
28 20 12 4
F x x x x x C
C.
1 1 1 1
( ) cos7 cos5 cos 3 cos
28 20 12 4
F x x x x x C
D.
1 1 1 1
( ) cos 7 cos5 cos 3 cos
28 20 12 4
F x x x x x C
Câu 140. Tìm nguyên hàm
Fx

( ) sinf x x x

A.
( ) 2 cos 4 sin 4 cosF x x x x x x C
B.
( ) 2 cos 4 sin 4 cosF x x x x x x C
C.
( ) 2 cos 4 sin 4 cosF x x x x x x C
D.
( ) 2 cos 4 sin 4 cosF x x x x x x C
Câu 141. Tính nguyên hàm
2
1x
xe dx

A.
2
1
1
()
2
x
F x e C
B.
2
1
1
()
2
x
F x e C
C.
2
1
1
()
2
x
F x e C
D.
2
1
()
2
x
F x e C
Câu 142. Tính
ln 2
2
x
dx
x

A.
( ) 2 2 1
x
F x C
B.
( ) 2 2 1
x
F x C
C.
( ) 2
x
F x C
D.
1
( ) 2
x
F x C
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Câu 143. 
2
1
()
1
fx
x
?
A.
2
()
1
x
Fx
x
B.
2
( ) ln 1F x x
C.
2
( ) ln 1F x x x
D.
2
( ) ln 1F x x x
Câu 144. 
2
()
1
x
fx
x
A.
2
1
( ) ln 1
2
F x x C
B.
2
( ) ln 1F x x C
C.
( ) 2F x x C
D.
11
( ) ln
21
x
F x C
x
Câu 145. Tìm
20
cos
sin
x
dx
x
.
A.
19
1
()
19sin
F x C
x
B.
19
1
()
19sin
F x C
x
C.
19
1
()
19cos
F x C
x
D.
19
1
()
19cos
F x C
x
Câu 146. 
( ) ln sin 3 cosF x x x

A.
cos 3 sin
()
sin 3 cos
xx
fx
xx
B.
cosx 3sin
()
sin 3 cos
x
fx
xx
C.
( ) cos 3sinf x x x
D.
sin 3 cos
()
cos 3sin
xx
fx
xx
Câu 147. Tìm
2
1
25
x
dx
xx
.
A.
2
22
25
x
C
xx
B.
2
2 2 5x x C
C.
2
25
2
xx
C
D.
2
25x x C
Câu 148. Nguyên hàm
Fx

23
( ) sin 2 .cos 2f x x x

0
2
F
A.
35
11
sin 2 sin 2
6 10
F x x x
B.
35
11
sin 2 sin 2
6 10
F x x x
C.
35
1 1 4
sin 2 sin 2
6 10 15
F x x x
D.
35
1 1 1
sin 2 sin 2
6 10 15
F x x x
Câu 149. 
()Fx

()
2
x
x
e
fx
e

0 ln 3F
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A.
( ) ln 2 ln 3
x
F x e
B.
( ) ln 2 ln 3
x
F x e
C.
( ) ln 2 2ln 3
x
F x e
D.
( ) ln 2 2ln 3
x
F x e
Câu 150. 
2
3
1
x
y f x
x
A.
33
1
3
xx
F x C
B.
3
1
3
x
F x C
C.
3
21
3
x
F x C
D.
3
21
3
x
F x C
Câu 151. 0034: 
A.
22
2
11
1
3
xx
x x dx C
B.
4
2
3
2
1
21
4
x
x x dx C
C.
4
2
3
22
1
2 1 .
4
x
x x dx x C
D.
2
2
ln 1
2
1
x
x
dx C
x
Câu 152. 
A.
2
ln
ln
x
dx x C
x
B.
2
ln ln
2
xx
dx C
x
C.
53
3 2 ln 3 2 ln
ln 3 2 ln
10 2
xx
xx
dx C
x
D.
23
ln ln
3
xx
xdx C
x
Câu 153. 
23
2
xx
y f x e x
A.
2 3 2
2
2 ln2 2
xx
ex
F x C
B.
2 3 2
2
2 ln 8 2
xx
ex
F x C
C.
32
2
2
ln 8 2
x
x
x
F x e C
D.
23
2
1
2 ln 8
xx
e
F x C
Câu 154. 
ln sin 3 cosF x x x


A.
cos 3 sin
sin 3 cos
xx
fx
xx
B.
cos 3sinf x x x
C.
cos 3 sin
sin 3 cos
xx
fx
xx
D.
sin 3 cos
cos 3sin
xx
fx
xx
Câu 155. Tính
ln 2
2.
x
dx
x

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A.
2 2 1
x
C
B.
2 2 1
x
C
C.
1
2
x
C
D.
2
x
C
Câu 156. 
A.
ln ln ln
ln .ln ln
dx
xC
x x x
B.
2
22
1 1 1
ln
2
1 1 1
dx x
C
x x x
C.
1
tan
1 cos 2 2
dx x
C
x
D.
2
2
1
ln 3 2
4
32
xdx
xC
x
Câu 157. 
3cos
( ) .sin
x
f x e x
A.
3cos
1
( ) .cos
3
x
f x dx e x C
B.
3cos
( ) 3
x
f x dx e C
C.
3 cos
1
()
3
x
f x dx e C
D.
3cos
( ) 3 .cos
x
f x dx e x C
Câu 158. 
ln
()
2
x
fx
x
A.
2
ln
()
4
x
f x dx C
B.
2
ln
()
2
x
f x dx C
C.
2
ln
()
4
x
f x dx C
x
D.
2
1
()
2
f x dx C
x
Câu 159. 
3
1.I x x dx
:
A. F(x) =
4 3 2
2 5 6 2
1 1 1 1 1
9 7 5 3
x x x x x C
B. F(x) =
4 3 2
2 6 6 2
1 1 1 1 1
9 7 5 3
x x x x x C
C. F(x) =
4 3 2
2 6 6 2
1 1 1 1 1
9 7 7 3
x x x x x C
D. F(x) =
4 3 2
2 6 6 1
1 1 1 1 1
9 7 5 3
x x x x x C
Câu 160. 
x
2x 1 4
d
I
:
A. F(x) =
2x 1 4 ln 2x 1 4 C
B. F(x) =
2x 1 4 ln 2x 1 4 C
C. F(x) =
2x 1 4 ln 2x 1 4 C
D. F(x) =
7
2x 1 ln 2x 1 4
2
C
Câu 161. 
5
cos
1 sin
x
y dx
x
là:
A.
34
sin cos
cos
34
xx
xC
B.
34
sin 3 cos 4
sin
34
xx
xC
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C.
34
sin cos
sin
34
xx
xC
D.
34
sin cos
sin
94
xx
xC
Câu 162. 
2
()
x
x
x x e
y dx
xe
:
A. F(x) =
1 ln 1
xx
xe xe C
B. F(x) =
1 ln 1
xx
e xe C
C. F(x) =
1 ln 1
xx
xe xe C
D. F(x) =
1 ln 1
xx
xe xe C
Câu 163. 
22
dx
y
xa
là:
A.
1
ln
2
xa
a x a
+C B.
1
ln
2
xa
a x a
+C C.
1
ln
xa
a x a
+C D.
1
ln
xa
a x a
+C
Câu 164. 
22
dx
y
ax
là:
A.
1
ln
2
ax
a a x
+C B.
1
ln
2
ax
a a x
+C C.
1
ln
xa
a x a
+C D.
1
ln
xa
a x a
+C
Câu 165. 
4 7 dxy x x
là:
A.
53
22
1 2 2
4 7 7 4 7
20 5 3
x x C
B.
53
22
1 2 2
4 7 7 4 7
18 5 3
x x C
C.
53
22
1 2 2
4 7 7 4 7
14 5 3
x x C
D.
53
22
1 2 2
4 7 7 4 7
16 5 3
x x C
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 T
ⓣⓗ
BÀI TP VN DNG
BT 13. 
a)
sinI x x dx
:
sin cos .I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
(1 2 )
x
I x e dx
:
(3 2 ) .
x
I x e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
cos
x
I e x dx
:
(sin cos ) .
2
x
e
I x x C
  
hay
:


: Suy ra:
u:  . u có
ln hay thì chn hay và còn li. Nu không
có thì chn c và còn l





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.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
(2 1) lnI x x dx
:
2
2
( )ln .
2
x
I x x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
3x
I x e dx
:
33
.
39
xx
xe e
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
2
ln2I x x dx
:
33
ln2
.
39
x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
lnI x dx
:
ln .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
h)
( 1) sin2I x x dx
:
11
cos2 sin2 .
24
x
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
x
I x e dx
:
(1 ) .
x
I x e C
.............................................................................................................................................................
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.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
j)
sin
x
I e x dx
:
(sin cos )
.
2
x
e x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
k)
cosI x x dx
:
sin cos .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
l)
sin
2
x
I x dx
:
2 cos 4 sin .
22
xx
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
m)
x
I x e dx
:
.
xx
I xe e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
n)
ln(1 )I x x dx
:
22
ln(1 ) (1 )
ln(1 ) .
2 2 4
x x x
I x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
o)
2
sinI x x dx
:
2
sin2 cos2
.
4 4 8
x x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
.............................................................................................................................................................
p)
2
ln( 1 )I x x dx
:
22
ln( 1 ) 1 .I x x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
q)
1
ln
1
x
I x dx
x
:
2
11
ln .
21
xx
I x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
r)
3
lnx
I dx
x
:
22
ln 1
.
24
x
IC
xx
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
s)
sin cosI x x x dx
:
11
cos2 sin2 .
48
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
t)
2
cos3
x
I e x dx
:
2
1
(3 sin 3 2 cos 3 ) .
13
x
I e x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
u)
1 cos2
x dx
I
x
:
11
tan ln cos .
22
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
.............................................................................................................................................................
v)
2
(2cos 1)I x x dx
:
1
sin2 cos2 .
24
x
I x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
w)
3
lnI x x dx
:
44
ln
.
4 16
x x x
IC
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
x)
2
sin
x
I dx
x
:
cot ln sin .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
y)
2
( 2)
x
I x e dx
:
22
11
( 2) .
24
xx
I x e e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
z)
2
ln( 1)I x x dx
:
2 2 2
( 1)ln( 1) 1 .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
BT 14. 
a)
2
2
1
ln
x
I x dx
x
:
11
ln .I x x x C
xx
.............................................................................................................................................................
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.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
b)
cosI x dx
:
2 sin 2 cos .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
c)
sinI x dx
:
2 cos 2 sin .I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
d)
2
3
(8 2 )
x
I x x e dx
:
22
2
(4 1) 4 .
xx
I x e e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
e)
2
3
.
x
I x e dx
:
22
2
11
.
22
xx
I x e e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
f)
3
5 x
I x e dx
:
33
3
11
.
33
xx
I x e e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
g)
sin
sin2
x
I e x dx
:
sin sin
2sin . 2 .
xx
I x e e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
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.............................................................................................................................................................
h)
x
I x e dx
:
2 4 4 .
x x x
I xe xe e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
i)
2
ln( 1)I x x dx
:
2 2 2
11
( 1)ln( 1) .
22
I x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
j)
2
1 ln( 1)x
I dx
x
:
11
ln 1 ln .
1
x
I x C
x x x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
k)
ln( 1)
xx
I e e dx
:
( 1)ln( 1) .
x x x
I e e e C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
l)
2
3
ln(4 8 3)
( 1)
xx
I dx
x
:
2
2
2
4 8 3
ln 4 8 3 4 ln 1 .
2( 1)
xx
x x x C
x
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
m)
1
1 ln( 1)
2
I x x dx
x
:
( 1)ln 1 .I x x x x x x C
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
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BÀI TP TRC NGHIM
Câu 166. 
()
x
f x xe
:
A.
x
eC
B.
1
x
e x C
C.
1
x
e x C
D.
2
2
x
x
eC
Câu 167. 
2
( ) ( 2 ).
x
f x x x e
:
A.
(2 2).
x
xe
B.
2 x
xe
C.
2
( ).
x
x x e
D.
2
( 2 ).
x
x x e
Câu 168. 
( ) .
x
f x x e

()Fx
()fx

(0) 1F
:
A.
( 1) 1
x
xe
B.
( 1) 2
x
xe
C.
( 1) 1
x
xe
D.
( 1) 2
x
xe
Câu 169. Cho
( ) sinf x x x

()fx
:
A.
cosx x C
B.
sin cosx x x C
C.
sin cosx x x C
D.
cos sinx x x C
Câu 170. 
2
()
x
f x xe

A.
2
( ) 2
x
F x e
B.
2
1
()
2
x
F x e
C.
2
2
( ) 2
x
F x x e
D.
22
()
xx
F x e xe
Câu 171. Cho
1
( ) ln
x
f x tdt

'( )fx

A.
1
x
B.
lnx
C.
2
ln x
D.
2
1
ln
2
x
Câu 172. 
3
()
x
f x x e
:
A.
32
( 3 6 6)
x
x x x e
B.
3
( 6 6)
x
x x e
C.
32
( 3 6)
x
x x e
D.
2
3
x
xe
Câu 173. 
2
.
x
f x x e
:
A.
2
2
x
F x e
B.
2
2
2
x
F x x e
C.
2
1
2
x
F x e
D.
22
xx
F x xe e
Câu 174. 
( ) ( 1)sinf x x x
có các nguyên hàm là:
A.
( ) ( 1)cos s inF x x x x C
B.
( ) ( 1)cos s inF x x x x C
C.
( ) ( 1)cos s inF x x x x C
D.
( ) ( 1)cos s inF x x x x C
Câu 175. 
( ) lnf x x
có các nguyên hàm là:
A.
( ) (ln 1)F x x x C
B.
1
()F x C
x
C.
2
ln
()
2
x
F x C
D.
( ) (ln 1)F x x x C
Câu 176. 
1
( ) cos
cos
f x x x
x
có các nguyên hàm là:
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A.
2
1
( ) sin
sin 2
x
F x x x C
x
B.
( ) (1 sin ) cosF x x x x C
C.
( ) (1 sin ) cosF x x x x C
D.
( ) (1 sin ) cosF x x x x C
Câu 177. 
()Fx

( ) cos 3f x x x

(0) 1F

()Fx
:
A.
11
( ) sin 3 cos 3
39
F x x x x C
B.
11
( ) sin 3 cos 3 1
39
F x x x x
C.
2
1
( ) sin 3
6
F x x x
D.
1 1 8
( ) sin 3 cos 3
3 9 9
F x x x x
Câu 178. Tính
x
xe dx

A.
xx
F x e xe C
B.
xx
F x e xe C
C.
xx
F x e xe C
D.
xx
F x e xe C
Câu 179. Tính
cosx xdx

A.
sin cosF x x x x C
B.
sin cosF x x x x C
C.
sin cosF x x x x C
D.
sin cosF x x x x C
Câu 180. Tìm
cos 2x xdx
:
A.
11
sin2 cos2
24
x x x C
B.
11
sin2 cos2
22
x x x C
C.
2
sin2
4
xx
C
D.
sin2xC
Câu 181. 
2
( ) 2
x
f x x x e
A.
( ) (2 2).
x
F x x e
B.
2
()
x
F x x e
C.
2
( ) ( ).
x
F x x x e
D.
2
( ) ( 2 ).
x
F x x x e
Câu 182. 
2
()
x
f x x e
A.
2
( ) ( 2 2)
x
F x x x e
B.
3
()
3
x
x
F x e
C.
2
( ) ( 2 2)
x
F x x x e
D.
( ) 2
x
F x xe
Câu 183. Nguyên hàm
Fx
()
x
f x xe

01F
A.
( ) 1 1
x
F x x e
B.
12
x
F x x e
C.
( ) 1 1
x
F x x e
D.
( ) 1 2
x
F x x e
Câu 184.  
A.
2
.cos
sin
2
xx
x xdx C
B.
sin cos sinx xdx x x x C
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ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 67 | THBTN
C.
cos sin cosx xdx x x x C
D.
cos 2 1
sin2 sin2
24
xx
x xdx x C
Câu 185. 
A.
3
33
1
39
x
xx
xe
xe dx e C
B.
x x x
xe dx xe e C
C.
2
.
2
xx
x
xe dx e C
D.
1
x x x
xx
dx C
e e e
Câu 186. 
A.
ln lnxdx x x x C
B.
1
lnxdx C
x
C.
22
ln ln
24
xx
x xdx x C
D.
33
2
ln .ln
39
xx
x xdx x C
Câu 187. 
A.
22
ln ln 2 lnxdx x x x x x C
B.
3
2
ln
ln
3
x
xdx C
C.
2
ln ln 1xx
dx C
xx
x
D.
3 2 2
ln ln 1
24
xx
dx C
x x x
Câu 188. 
A.
2 2 2
1
24
x x x
xx
dx C
e e e
B.
x x x
xe dx xe e C
C.
3
33
1
39
x
xx
xe
xe dx e C
D.
2
22
.
2
xx
x
xe dx e C
Câu 189. 
A.
3
2
1
ln .
3
x
x xdx C
x
B.
33
2
ln .ln
39
xx
x xdx x C
C.
2 2 2
ln 1 ln 1 1x x dx x x x x C
D.
sin cos
sin
2
x
x
e x x
e xdx C
Câu 190. 
( ) .sin 2 1f x x x
A.
1
( ) .cos 2 1 .sin 2 1
24
x
f x dx x x C
B.
2
( ) .cos 2 1
4
x
f x dx x C
C.
1
( ) .cos 2 1 .sin 2 1
24
x
f x dx x x C
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D.
1
( ) .cos 2 1 .sin 2 1
22
x
f x dx x x C
Câu 191. 
( ) .ln 1f x x x
A.
2
()
2( 1)
x
f x dx C
x
B.
2
3
1
( ) ln 1 ln(1 )
26
x
f x dx x x x C
C.
22
11
( ) 1 .ln 1
2 4 2
x
f x dx x x x C
D.
2
2
11
( ) ln 1 ln( 1)
2 4 2 2
xx
f x dx x x x C
Câu 192. 
cos2 .ln(sin cos )I x x x dx
:
A. F(x) =
11
1 sin2 ln 1 sin2 sin2
24
x x x C
B. F(x) =
11
1 sin2 ln 1 sin2 sin2
42
x x x C
C. F(x) =
11
1 sin2 ln 1 sin2 sin2
44
x x x C
D. F(x) =
11
1 sin2 ln 1 sin2 sin2
44
x x x C
Câu 193. 
2 sin 3I x xdx
:
A. F(x) =
2 cos 3
1
sin 3
39
xx
xC
B. F(x) =
2 cos 3
1
sin 3
39
xx
xC
C. F(x) =
2 cos 3
1
sin 3
39
xx
xC
D. F(x) =
2 cos 3
1
sin 3
33
xx
xC
BÀI TP TRC NGHIM DNG TOÁN KHÁC C THÊM )
Câu 194. 
A. m s
2
61
()
23
xx
Fx
x
và
2
10
()
23
x
Gx
x
là nguyên hàm ca cùng mt hàm
s.
B. m s
2
( ) 5 2 sinF x x
và
( ) 1 cos2G x x
nguyên hàm ca cùng mt hàm
s.
C. Hàm s
2
( ) 2 2F x x x
nguyên hàm ca hàm s
2
1
()
22
x
fx
xx
.
D. Hàm s
( ) sinF x x
ngun hàm ca hàm s
( ) cosf x x
.
Câu 195. SAI?
TÀI LIU HC TP TOÁN 12 CHẤT LƯỢNG BÀI GING: NGUYÊN HÀM TÍCH PHÂN
ADMIN TRẦN VĂN TÀI 0977.413.341 - TOÁN HC BC TRUNG NAM 69 | THBTN
A.
kf x dx k f x dx k R
B.
..f x g x dx f x dx g x dx
C.
f x g x dx f x dx g x dx
D.
1
1
m
m
fx
f x f x dx C
m
Câu 196. 
sin cos
x
F x a x b x e

cos .
x
f x x e

a
,
b
:
A.
1, 0ab
B.
0, 1ab
C.
1ab
D.
1
2
ab
Câu 197. 
fx

K

Fx

fx
trên
K
A.
,F x f x x K
B.
,F x f x x K
C.
,f x F x x K
D.
,f x F x x K
Câu 198. sai?
A.
'( ) ( )f x dx f x C
B.
( ) ( ) 0Kf x dx K f x dx K
C.
( ) ( ) ( ) ( )f x g x dx f x dx g x dx
D.
( ) ( )F x dx f x C
Câu 199. 
A.
3
2
()
'( ) ( )
3
fx
f x f x dx C
B.
( ). ( ) ( ) . ( )f x g x dx f x dx g x dx
C.
( ) ( ) ( ) ( )f x g x dx f x dx g x dx
D.
( ) ( )kf x dx k f x dx
(
k
hng s)
Câu 200. 
2
( ) ( ) 2 1f x ax bx c x

2
10 7 2
()
21
xx
gx
x

1
;
2
thì
a b c

A. 3 B. 0 C. 4 D. 2
Câu 201. 
a
,
b
,
c
sao cho
2
( ) ( ) 2 3g x ax bx c x

2
20 30 7
()
23
xx
fx
x

3
;
2
A.
4, 2, 2a b c
B.
1, 2, 4a b c
C.
2, 1, 4a b c
D.
4, 2, 1a b c
Câu 202. Tìm nguyên hàm
22
( ) ( tan tan )
x
F x e a x b x c
   
23
( ) tan
x
f x e x

;
22
TÀI LIU HC TP TOÁN 12 BÀI GING: NGUYÊN HÀM TÍCH PHÂN
70 | THBT CA GIÁO VIÊN CN FILE WORD LIÊN H THY TÀI: 0977.413.341
A.
22
1 2 2
( ) tan tan
2 2 2
x
F x e x x
B.
22
1 2 1
( ) tan tan
2 2 2
x
F x e x x
C.
22
1 2 1
( ) tan tan
2 2 2
x
F x e x x
D.
22
1 2 2
( ) tan tan
2 2 2
x
F x e x x
Câu 203. 
2
()
x
F x ax bx c e
    
2
2 7 4
x
f x x x e
thì
;;a b c

A.
2; 3;1
B.
1;3;2
C.
1; 1;1
D.
Câu 204. Cho
f x dx F x C

0a
, ta
f ax b dx

A.
1
2
F ax b C
a
B.
1
F ax b C
a
C.
F ax b C
D.
.a F ax b C
Câu 205. Cho
sin
cos . ; 0
1
;0
1
x
x e x
fx
x
x

A.
sin
;0
2 1 1; 0
x
ex
Fx
xx
là mt nguyên hàm ca
fx
B.
cos
;0
2 1 1; 0
x
ex
Fx
xx
là mt nguyên hàm ca
fx
C.
1 cos
;0
2 1 ; 0
x
ex
Fx
xx
là mt ngun hàm ca
fx
D.
cos
;0
2 1 ; 0
x
ex
Fx
xx
là mt nguyên hàm ca
fx
Câu 206.    
cos 3
sin 3
2 sin 3 2017
x a x
x
x xdx
bc
 
.S a b c

A.
15S
B.
14S
C.
3S
D.
10S
Câu 207.    
2
2
20 30 7
; 2 3
23
xx
f x F x ax bx c x
x
3
2
x


Fx

fx

,,a b c
A.
4; 2; 1a b c
B.
4; 2; 1a b c
C.
4; 2; 1a b c
D.
4; 2; 1a b c
| 1/70

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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
PHẦN 1: NGUYÊN HÀM VÀ PHƢƠNG PHÁP TÌM NGUYÊN HÀM ⓣⓗⓑⓣⓝ
Khái niệm nguyên hàm và tính chất
1. Khái niệm nguyên hàm
— h h f (x) h K. H F(x) g i nguyên hàm h
f (x) trên K F (x) f (x), x K.
— N F(x) g h f (x) trên K h h nguyên hàm h f (x) trên K f (x) dx
F(x) C, const C .
2. Tính chất N f (x), g(x) 2 h i ụ K k 0 thì ta luôn có: f (x)dx
f (x) C. kf (x)dx k f (x)dx. f (x) g(x)dx f (x)dx g(x)dx
ảng nguyên hàm c m t hàm thƣ ng g p v i C à h ng t y 1 x n 1 ax b n 1 ( )  x dx C (ax b) dx C 1 a n 1 1 1 1  dx ln x C dx ln ax b C x ax b a 1 1 1 1 1  dx C dx C 2 x x 2 (ax b) a ax b  sin x dx cosx C 1 sin(ax b)dx cos(ax b) C a  cosx dx sin x C 1 cos(ax b) dx sin(ax b) C a 1 1 1  dx cotx C dx cot(ax b) C 2 sin x 2 sin (ax b) a 1 1 1  dx tan x C dx tan(ax b) C 2 cos x 2 cos (ax b) ax x e dx e C ax b 1 ax b e dx e C a x a dx 1 x ax a dx C ln C lna 2 2 x a 2a x a 1
♦ Nhận xét. Khi thay x g (ax b) h g h h h a
ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC – TRUNG – NAM 1 | THBTN
TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN M t ƣu
1. ầ ắ vữ g g g h .
2. Ng h h h g hi h h g
h h g g h hữ g h h h hầ .
3. M g h h h i h
hữ g h g h v g g h .
Dạng toán 1. TÍNH NGUYÊN HÀM ẰNG ẢNG NGUYÊN HÀM ⓣⓗⓑⓣⓝ Phƣơng Pháp 1. T h hứ h ặ ũ hừ h i iể . 2. T h h ũ h i iể he ô g hứ ũ. 3. hứ ă h ể v ũ hừ . 4. T h g gi ậ i v i h i iể he ô g hứ h h h ổ g.
5. Bậc chẵn c a sin và cosin Hạ bậc.
BÀI TẬP VẬN DỤNG Bài 1
Tìm nguyên hàm c a các hàm s gi ử i iệ h
P ươ p áp: Dựa vào bảng nguyên hàm của các hàm số và vận dụng các tính chất nguyên hàm. a 2 a a) 2 f (a) 3a ĐS: 3 F(a) a C. 2 4
………………………………………………………………………………………………… 4 2 b 5b b) 3 f (b) 2b 5b 7. ĐS: F(b) 7b C. 2 2
2 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
………………………………………………………………………………………………… 3 c c) 5 3 2 f (c) 6c 12c c 8. ĐS: 6 4 F(x) c 3c 8c C. 3
………………………………………………………………………………………………… 4 3 2 x 2x 3x d) 2 f (x) (x 3x) (x 1) ĐS: F(x) C. 4 3 2
………………………………………………………………………………………………… 4 (3 x) e) 3 f (x) (3 x) . ĐS: F(x) C. 4
………………………………………………………………………………………………… 1 1 3 1 x x f) 2 f (x) x ĐS: F(x) C. 2 x 3 x 3 3
………………………………………………………………………………………………… 2 10 x g) 2 ( ) 10 x f x . ĐS: F(x) C. 2 ln10
………………………………………………………………………………………………… 3 4 x h) 3 f (x) x 4x ĐS: 2 F(x) 2x 3.ln x C. x 4
………………………………………………………………………………………………… 4 2t 2 2 2 i) f (t) ĐS: 3 F(t) t C. t 3 t
………………………………………………………………………………………………… x 1 1 j) f (x) ĐS: F(x) ln x C. 2 x x
………………………………………………………………………………………………… x k) 2 f (x) 2 sin ĐS: F(x) x sin x C. 2
………………………………………………………………………………………………… 1 1 l) 2 f (x) cos x. ĐS: F(x) x sin 2x C. 2 4
………………………………………………………………………………………………… m) 2 f (x) tan x. ĐS: F(x) tan x x C.
………………………………………………………………………………………………… 1 n) f (x) ĐS: F(x) 2cot2x C. 2 2 sin x.cos x
…………………………………………………………………………………………………
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1 o) f (x) 2 sin 3x cos2x. ĐS: F(x) cos 5x cos x C. 5
………………………………………………………………………………………………… 1 p) ( ) x .( x f x e e 1). ĐS: 2 ( ) x x F x e e C. 2
………………………………………………………………………………………………… x e q) f (x) x e 2 ĐS: ( ) 2 x F x e tan x C. 2 cos x
………………………………………………………………………………………………… 2 3 r) 3 I ( x x ) dx. ĐS: 3 I x C. 2
………………………………………………………………………………………………… 1 2 s) 2 I 2x dx ĐS: 3 3 I x 3 x C. 3 2 x 3
………………………………………………………………………………………………… 1 3 5 9 25 t) I dx ĐS: 3 2 5 4 F(x) x x x C. 3 5 2 x x x 2 4
………………………………………………………………………………………………… u) 2 I 4 sin x dx. ĐS: I 2x sin2x C.
………………………………………………………………………………………………… 1 cos 4x x sin 4x v) I dx. ĐS: I C. 2 2 8
………………………………………………………………………………………………… x 1 3 w) x 1 I (3 cosx 3 ) dx ĐS: I 3 sin x C. ln 3
………………………………………………………………………………………………… x) 2 I (tan x 2 cotx) .dx. ĐS: I tanx 4 cotx 9x C.
………………………………………………………………………………………………… 3 y) 3 I u.(u 4).d . u ĐS: 3 7 3 4 I u 3 u C. 7
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
4 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN Bài 2 Chứng minh
là m t nguyên hàm c a hàm s g ờng h p sau:
P ươ p áp: Để
là một nguyên hàm của hàm số
ta cần chứng minh: a) 3 2 F(x) 5x 4x 7x 120 và 2 f (x) 15x 8x 7.
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 1 b) 2 F(x) ln(x x 3)và f (x) . 2 x 3
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. c) ( ) (4 5) x F x x e và ( ) (4 1) x f x x e .
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. d) 4 F(x) tan x 3x 5 và 5 3 f (x) 4 tan x 4 tan x 3.
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 x 4 2x e) F(x) ln và f (x) . 2 x 3 2 2 (x 4) (x 3)
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 x x 2 1 2 2 2(x 1) f) F(x) ln và f (x) . 2 x x 2 1 4 x 1
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN Bài 3
Tìm nguyên hàm c a các hàm s thỏ ã i u kiệ h ớ g ờng h p sau:
P ươ p áp: Tìm nguyên hàm của hàm số tức đi tính Rồi sau đó thế
để tìm hằng số 4 x 5 a) 3 f (x) x 4x 5, F(1) 3. ĐS: 2 F(x) x 5x 4 4
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. b) f (x) 3 5 cosx, F( ) 2. ĐS: F(x) 3x 5sin x 2 3 .
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 3 5x 2 2 5x 5e c) f (x) , F(e) 1. ĐS: F(x) 3 ln x 2. x 2 2
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 x 1 3 2 x d) f (x) , F(1) ĐS: F(x) ln x 1. x 2 2
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 1 2 22 e) f (x) x x , F(1) 2. ĐS: 5 F(x) x 2 x x 5 5
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 1 1 1 f) I
sin 2x.cos x.dx, i F 0. ĐS: F(x) cos 3x cos x 3 6 2 12
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
6 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 4 3 3x 2x 5 5 g) I dx, i F(1) 2. ĐS: 3 2 F(x) x x 7. 2 x x
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 3 2 x 3x 3x 7 2 x 8 h) I dx, i F(0) 8. ĐS: F(x) x 2 (x 1) 2 x 1
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. x x sin x 1 i) 2 I sin dx, i F ĐS: F(x) 2 2 4 2 2 2
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 1 7 3 x 13 j) I x x dx, i F(1) ĐS: F(x) x . x 2 3 6
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 2 cos x 1 k) I dx, i F ĐS: F(x) 2x tan x 1. 2 cos x 4 2
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. Bài 3
T i u kiện c a tham s m hoặc a, b, c
là m t nguyên hàm c a hàm s
P ươ p áp: Để
là một nguyên hàm của hàm số Từ
đó, ta sử dụng đồng nhất thức để tìm ra tham số cần tìm. 3 2 F(x) mx (3m 2)x 4x 3 a) ĐS: m 1. 2 f (x) 3x 10x 4
.............................................................................................................................................................
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 F(x) ln x mx 5 b) 2x 3 ĐS: m 3. f (x) 2 x 3x 5
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 F(x) (ax bx c) x e c) ĐS: a 0, b 1, c 4. f (x) (x 3) x e
.............................................................................................................................................................
.............................................................................................................................................................
.............................................................................................................................................................
............................................................................................................................................................. 2 2 F(x) (ax bx c) x e d) ĐS: a 1, b 3, c 2. 2 2 f (x) (2x 8x 7) x e
.............................................................................................................................................................
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............................................................................................................................................................. 2 F(x) (ax bx c) x e e) ĐS: a 1, b 1, c 1. 2 f (x) (x 3x 2) x e
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............................................................................................................................................................. b c F(x) (a 1)sin x sin 2x sin 3x f) 2 3 ĐS: a b c 0. f (x) cos x
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 F(x) (ax bx c) 2x 3 g) 2 20x 30x 7 ĐS: a 4, b 2, c 1. f (x) 2x 3
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............................................................................................................................................................. f (x) x 3 x, (x 3) 2 2 12 h) ĐS: a ; b ; c 2 F(x) (ax bx c) 3 x 5 5 5
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BÀI TẬP TRẮC NGHIỆM
NHÓM 1 : DÙNG BẢNG NGUYÊN HÀM Câu 1. Ng h h 3 f x x 3x 2 h g h ? 4 2 x 3x 4 x A. F x 2x C . B. 2 F x 3x 2x C . 4 2 3 4 2 x x C. F x 2x C . D. 2 F x 3x 3x C . 4 2 Câu 2. H 3 2 F x 5x 4x 7x 120 C h g h h ? A. 2 f x 15x 8x 7 . B. 2 f x 5x 4x 7 . 2 3 2 5x 4x 7x C. f x . D. 2 f x 5x 4x 7 . 4 3 2 1 Câu 3. H g h h 2 y x 3xx 3 x 3 3 x 3 A. 2 F x x ln x C . B. 2 F x x ln x C . 3 2 3 2 3 x 3 1 C. 2 F x x ln x C . D. F x 2x 3 C . 3 2 2 x
Câu 4. T g h h f x x 1 x 2 3 x 3 3 x 2 A. 2 F x x 2x C . B. 2 F x x 2x C . 3 2 3 3 3 x 2 C. F x 2x 3 C . D. 2 F x x 2x C . 3 3
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 2 3
Câu 5. Nguyên hàm F x h f x h ? 2 5 2x x x 3 3 A. F x ln 5 2x 2 ln x C . B. F x ln 5 2x 2 ln x C . x x 3 3 C. F x ln 5 2x 2 ln x C . D. F x ln 5 2x 2 ln x C . x x Câu 6. M g h h 3 2 f (x) x 3x 5 là 4 x A. 2 3x 6x B. 2 3x 6x C C. 3 x 5x C D. 4 3 x x 5x C 4 4
Câu 7. M g h h f (x) x 3 là 4 5 3 x 3 3 x 3 x 3 A. B. 4 x 3 C. D. 4 5 3 Câu 8. M g h h 3 2 f (x) x 3x 5 là 4 x A. 2 3x 6x B. 2 3x 6x C C. 3 x 5x C D. 4 3 x x 5x C 4 Câu 9. M g h h 4 2 g(x) 5x 4x 6 là: 4 4 A. 5 3 x x 6x C B. 3 20x 8x C C. 3 20x 8x D. 5 3 x x C 3 3 1 Câu 10. Tính 2 3x 2 dx x 3 x 1 A. ln x 2x C B. 3 x 2x C 3 2 x C. 3 x ln x C D. 3 x ln x 2x C Câu 11. Cho 3 2 f (x) x 3x
2x . M g h F(x) f (x) hỏ F 1 0 là: 4 x 1 4 x 1 A. 3 2 x x B. 3 2 x x 4 4 4 4 4 x 4 x C. 3 2 x x 1 D. 3 2 x x 1 4 4
Câu 12. G i F(x) ậ h g h h 2 f (x) 3x 2x
1 thì F(x) là: A. 3 2 F(x) 3x 2x x C B. 3 2 F(x) x x 1 C 1 1 C. 2 F(x) x(x x 1) C D. 3 2 F(x) x x x C 3 2 2 Câu 13. K 2 x x 1 dx g 3 3 2 x 1 2 x 1 A. F(x) C B. F(x) C 3 6
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 3 x x 2 3 x C. F(x) x C D. 2 F(x) x 1 C 2 3 6 1
Câu 14. M g h F x h f (x) là: x 1 1 1 A. F(x) B. F(x) C. F(x) ln x D. F(x) 2 x 2 x x 2
Câu 15. T h g h F x h f (x) 3 sin x x A. F(x) 3 cosx 2 ln x C B. F(x) 3 cosx 2 ln x C C. F(x) 3 cosx 2 ln x C D. F(x) 3 cosx 2 ln x C
Câu 16. T h g h F x h 2 3 – 3x f x x 3x 3x A. 3 F(x) x C B. 3 F(x) x C ln 3 ln 3 3 3x x 3 3x x C. F(x) C D. F(x) C 3 ln 3 3 ln 3 9 Câu 17. Tính 1 x dx 1 10 1 10 A. F(x) 1 x C B. F(x) 1 x C 10 10 1 10 1 10 C. F(x) 1 x C D. F(x) 1 x C 10 10 2
Câu 18. Tìm nguyên hàm F x h f x x x 2 4 x 4 4 x A. 3 2 F(x) x 2x C B. 2 F(x) 2x C 4 3 4 3 x C. 2 F(x) 2x C D. K t qu khác. 3 Câu 19. H g h 2 f (x) x 2x 1 là: 1 A. 3 F(x) x 2 x C B. F(x) 2x 2 C 3 1 1 C. 3 2 F(x) x x x C D. 3 2 F(x) x 2x x C 3 3 Câu 20. Ng h h 3 f (x) (2x 1) là: 1 A. 4 (2x 1) C B. 4 (2x 1) C C. 4 2(2x 1) C D. K t qu khác 2 Câu 21. Ng h h 5 f (x) (1 2x) là: 1 A. 6 (1 2x) C B. 6 (1 2x) C C. 6 5(1 2x) C D. 4 5(1 2x) C 12
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3
Câu 22. Ng h h f = 2x là : 2 x 3 3 A. 2 x C B. 2 x C C. 2 2 x 3 lnx
C D. K t qu khác x 2 x
Câu 23. T h f x i g f x 2x 1 và f 1 5 A. 2 x x 3 B. 2 x x 3 C. 2 x x D. K t qu khác Câu 24. T h y f (x) i 2 f (x) (x x)(x 1) và f(0) 3 4 2 x x 4 2 x x A. y f (x) 3 B. y f (x) 3 4 2 4 2 4 2 x x C. y f (x) 3 D. 2 y f (x) 3x 1 4 2 Câu 25. Cho 2 f (x) 3x 2x 3 ó g h iệ i hi x 1 . Ng h ó ? A. 3 2 F(x) x x 3x B. 3 2 F(x) x x 3x 1 C. 3 2 F(x) x x 3x 2 D. 3 2 F(x) x x 3x 1 b
Câu 26. T h f x i g f '(x) ax , f '(1) 0, f (1) 4, f ( 1) 2 2 x 2 x 1 5 2 x 1 5 2 x 1 5 A. B. C. D. K t qu khác 2 x 2 2 x 2 2 x 2 3 Câu 27. M g h h 2 f x x 2 x x 3 x 4 3 x 3 A. 3 F x 3 ln x x C. 3 F x 3ln x x 3 3 3 4 3 x 3 4 3 x 4 B. 3 F x x D. 3 F x 3 ln x x 3 x 3 3 3
Câu 28. Nguyên hàm F x 3 2 f x x 3x 2x hỏ ã F 1 0 là: 4 x 1 4 x 1 A. 3 2 F x x x B. 3 2 F x x x 4 4 4 4 4 x 4 x C. 3 2 F x x x 1 D. 3 2 F x x x 1 4 4
NHÓM 2: HÀM SỐ VÔ TỶ ( CHỨA CĂN 1
Câu 29. Ng h h f (x) là 2x 1 A. f x x d 2x 1 C . B. f x x d 2 2x 1 C . 2x 1 C. f x x d C . D. f x x d 2 2x 1 C . 2
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1
Câu 30. T g h h f (x) . 3 x A. f x x d 2 3 x C . B. f x x d 3 x C . C. f x x d 2 3 x C . D. f x x d 3 3 x C .
Câu 31. T g h h f (x) 2x 1 . 1 2 A. f x x d 2x 1 2x 1 C . B. f x x d 2x 1 2x 1 C . 3 3 1 1 C. f x x d 2x 1 C . D. f x x d 2x 1 C . 3 2
Câu 32. T g h h f (x) 5 3x . 2 2 A. f x x d 5 3x 5 3x C . B. f x x d 5 3x 5 3x . 9 3 2 2 C. f x x d 5 3x 5 3x . D. f x x d 5 3x C . 9 3 Câu 33. T g h h 3 f (x) x 2 . 3 3 A. 3 f x x d x 2 x 2 C . B. 3 f x x d x 2 x 2 C . 4 4 2 2 1 C. f x x d x 2 x 2 . D. 3 f x x d x 2 C . 3 3 Câu 34. T g h h 3 f (x) 1 3x . 1 3 A. 3 f x x d 1 3x 1 3x C . B. 3 f x x d 1 3x 1 3x C . 4 4 1 2 C. 3 f x x d 1 3x 1 3x C . D. 3 f x x d 1 3x C . 4 2 Câu 35. H F x x 1 x 1 2016 g h h ? 5 5 A. f x x 1 x 1 B. f x x 1 x 1 C 2 2 2 C. f x x 1 x 1 D. f x x 1 x 1 C 5 1
Câu 36. Bi g h h f x 1 h F x hỏ ã 1 3x 2 F 1 . Khi ó F x h ? 3 2 2 A. F x x 1 3x 3 B. F x x 1 3x 3 3 3 2 2 C. F x x 1 3x 1 D. F x 4 1 3x 3 3
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN a
Câu 37. Bi F(x)
6 1 x g h h f (x) . Khi ó gi 1 x a g 1 A. 3 . B. 3 . C. 6 . D. . 6 1 1 Câu 38. Tính dx x 2 x x x 1 1 2 x A. C B. 2 x C C. x C D. C 2 2 2 2 x 2 x 2 1
Câu 39. h h f x 1 . Khi ó x 1 A. f x dx C B. f x dx x ln x C 2 x 2 1 1 1 C. f x dx x C D. f x dx 1 2 x 2 x 1
Câu 40. G i F(x) ậ h g h h f (x) thì F(x) là: 1 2x 1 1 A. F(x) ln 1 2x C B. F(x) ln 1 2x C 2 2 x C. F(x) ln 1 2x C D. F(x) C 2 x x
Câu 41. T g h F h f (x) 3x 4 i F(0) 2 . K 2 3 2 2 3 2 A. F(x) 3x 4 B. F(x) 3x 4 9 9 9 9 2 3 10 2 3 10 C. F(x) 3x 4 D. F(x) 3x 4 3 3 3 3 4
Câu 42. Tìm nguyên hàm 3 2 x dx x 5 3 A. 3 5 x 4 ln x C B. 3 5 x 4 ln x C 3 5 3 3 C. 3 5 x 4 ln x C D. 3 5 x 4 ln x C 5 5 Câu 43. H g h h 2 f (x) x k với k 0? x k 1 x A. 2 2 f (x) x k ln x x k B. 2 2 f (x) x k ln x x k 2 2 2 2 k 1 C. 2 f (x) ln x x k D. f (x) 2 2 x k
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN Câu 44. T g h (I) 2 f (x) x 1 (II) 2 f (x) x 1 5 1 1 (III) f (x) (IV) f (x) - 2 2 x 1 2 x 1
Hàm s nào có m t nguyên hàm là hàm s 2 F(x) ln x x 1 A. Chỉ (I) B. Chỉ (III) C. Chỉ (II)
D. Chỉ (III) và (IV) 2 Câu 45. M g h h 3 1 f (x) x h x 3 3 12 1 1 A. 3 2 6 5 F(x) x x x ln x B. 3 F(x) x 5 5 3 x 2 3 12 C. 3 F(x) x x x D. 3 2 5 6 F(x) x x ln x x 5 5 Câu 46. Nguyên hàm 2017x x x e dx = 2017 5 x e 2017 2 x e A. 2 x x C B. 3 x x C 2 2017 5 2017 2017 3 x e 2017 2 x e C. 2 x x C D. 2 x x C 5 2017 5 2017
NHÓM 3: HÀM SỐ LƢỢNG GIÁC
Câu 47. T g h h f (x) cos 3x . 6 1 A. f (x)dx sin 3x C . B. f (x).dx sin 3x C . 3 6 6 1 1 C. f (x)dx sin 3x C . D. f (x)dx sin 3x C . 3 6 6 6 x Câu 48. T g h h 2 f (x) 1 a t n . 2 x x A. f (x)dx 2 tan C . B. f (x)dx tan C . 2 2 1 x x C. f (x)dx tan C . D. f (x)dx 2 tan C . 2 2 2 1
Câu 49. T g h h f (x) . 2 sin x 3 1 A. f (x)dx cot x C . B. f (x)dx cot x C . 3 3 3
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1 C. f (x)dx cot x C . D. f (x)dx cot x C . 3 3 3 Câu 50. Tính sin x cosx dx A. cosx sinx C B. cosx sinx C C. cosx sinx C D. cosx sinx C 2
Câu 51. M g h h f (x) là: 2 cos x A. 2 tan x C B. 2 cotx C C. 2 sin x C D. 2 cos x C 1
Câu 52. M g h h f (x) 3 là: 2 sin x A. 3x tanx C B. 3x tanx C C. 3x cotx C D. 3x cotx C
Câu 53. Cho f (x) sinx
cosx . M g h F(x) f (x) hỏ F 0 là: 4 2 A. cos x sinx 2 B. cos x sin x 2 2 C. cos x sin x 2 D. cos x sin x 2
Câu 54. h h f (x) 2x sinx
2 cosx . M g h F(x) f (x) hỏ F(0) 1 là: A. 2 x cosx 2 sinx 2 B. 2 x cosx 2 sinx 2 C. 2 cosx 2sinx D. 2 x cosx 2 sinx 2 Câu 55. M g h h 2 f (x) tan x là: 3 tan x 3 tan x 1 2 sin x A. B. . C. tanx x D. 3 2 3 cos x 3 cos x Câu 56. M g h h 4 4 f (x) cos x sin x là: 1 A. cos 2x B. sin 2x C. 2 sin 2x D. 2 cos x 2 Câu 57. H ới g h h 4 4 f (x) sin x cos x ? 1 1 3 1 3 1 A. x sin 4x B. x sin 4x C. x sin 4x D. x cos 4x 4 4 4 16 4 4 Câu 58. M g h h 2 f x sin 2x 3x là: 1 A. F x cos2x 6x B. F x cos 2x 6x 2 1 1 C. 3 F x cos 2x x D. 3 F x cos 2x x 2 2
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
Câu 59. H g h h f x 2 sin 2x ? A. 2 F x sin x B. F x 2 cos2x 1 C. F x cos 2x D. F x cos2x 2
Câu 60. H f (x) sinx ó g h A. F(x) cosx C B. F(x) sinx C C. F(x) cosx 1 D. F(x) sinx C Câu 61. Bi 2 F(x) 1
tan x dx hi ó F(x) là: 1 A. F(x) C B. F(x) tan x C 2 cos x C. F(x) tanx C D. F(x) cotx C
Câu 62. G i F(x) ậ h g h h f (x)
sin2x thì F(x) là: 1 A. F(x) cos 2x C B. F(x) 2 sin2x C 2 1 C. F(x) cos 2x C D. F(x) 2 sin2x C 2
Câu 63. G i F (x) g h 2 f (x)
sin x hỏ ã F (0)
0 và F (x) là nguyên 1 1 1 2 h 2 f (x)
cos x hỏ ã F (0)
0 . Khi ó h g h F (x) F (x) có 2 2 1 2 ghiệ A. x k ,k Z B. x k,k Z C. x k ,k Z D. x k2 ,k Z 2 2 Câu 64. Ng h h 2 y
cos x.sinx là: 1 1 A. 3 cos x C B. 3 cos x C C. 3 sin x C D. Đ h . 3 3
Câu 65. M g h h y
cos 5x.cosx là: A. F x cos 6x B. F x sin 6x 1 1 1 1 sin 6x sin 4x C. sin 6x sin 4x D. 2 6 4 2 6 4
Câu 66. M g h h y
sin 5x.cos 3x là: 1 cos 6x cos 2x 1 cos 8x cos 2x A. B. 2 8 2 2 8 2 C. cos 8x cos2x D. Đ h .
Câu 67. T h g h F x h f x cosx A. F x sin x C B. F x sin x C
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN C. F x cosx C D. F x cosx C
Câu 68. K g h h f x cosx i g h iệ i hi x ? 2 A. F x sin x B. F x sin x C. F x sin x 1 D. F x sin x 1 1 Câu 69. Tính dx 2 cos 3x 1 A. F x tan 3x – 1 C B. F x cot 3x – 1 C 1 1 C. F x tan 3x – 1 C D. F x cot 3x – 1 C 3 3 Câu 70. T g h F h 2 f x tan x i F 1. K 4 A. F(x) tan x x B. F(x) tan x x 4 4 C. F(x) x tan x D. F(x) x tan x 4 4 Câu 71. Tính sin(3x 1)dx 1 1 A. cos(3x
1) C B. cos(3x
1) C C. cos(3x
1) C D. K t qu khác 3 3 Câu 72. Tìm (cos 6x cos 4x)dx là: 1 1 A. sin 6x sin 4x C B. 6 sin 6x 5sin 4x C 6 4 1 1 C. sin 6x sin 4x C D. 6 sin 6x sin 4x C 6 4 Câu 73. T g h 2 (I) 2 f (x) tan x 2 (II) f (x) (III) 2 f (x) tan x 1 2 cos x
Hàm s nào có m t nguyên hàm là hàm s g(x) = tanx A. (I), (II), (III) B. Chỉ (II), (III) C. Chỉ (III) D. Chỉ (II)
Câu 74. Ng h h f x
2 sin 3x cos2x 1 1 A. cos 5x cos x C B. cos 5x cos x C 5 5 C. 5 cos 5x cosx C D. K t qu khác
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN Câu 75. L h h g ú g A. cotxdx ln sin x C B. sin xdx cosx C 1 1 C. dx C D. cos xdx sin x C 2 x x
Câu 76. Tìm nguyên hàm 2 (1 sin x) dx 2 1 2 1 A. x 2 cos x sin 2x C B. x 2 cos x sin 2x C 3 4 3 4 2 1 3 1 C. x 2 cos 2x sin 2x C D. x 2 cos x sin 2x C 3 4 2 4
Câu 77. Trong ệ h ệ h i ? 1 1
(I ) sin x sin 3xdx
(sin 2x - sin 4x) C 4 2 1 2 3 (II ) tan xdx tan x C 3 x 1 1 2 (III ) dx ln(x 2x 3) C 2 x 2x 3 2 A. Chỉ (I) và (II) B. Chỉ (III)
C. Chỉ (II) và (III) D. Chỉ (II) Câu 78. Tìm 3 (sin x 1) cos xdx là: 4 (cos x 1) 4 sin x 4 (sin x 1) A. C B. C C. C D. 3 4(sinx 1) C 4 4 4 Câu 79. Xé ệ h 2 x x (I) F(x) x
cosx là m t nguyên hàm c a f (x) sin - cos 2 2 4 x 3 (II) F(x)
6 x là m t nguyên hàm c a 3 f (x) x 4 x (III) F(x)
tanx là m t nguyên hàm c a f(x) - ln cosx Mệ h nào sai ? A. (I) và (II) B. Chỉ (III) C. Chỉ (II) D. Chỉ (I) và (III) 3x Câu 80. Tìm sin dx . 2 2 3x 2 3x A. F(x) cos C B. F(x) cos C 3 2 3 2 3 3x 3 3x C. F(x) cos C D. F(x) cos C 2 2 2 2 Câu 81. Ng h h 3 y sin x.cosx là: 1 1 A. 4 F(x) sin x C B. 4 F(x) sin x C 4 4
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1 1 C. 4 F(x) cos x C D. 4 F(x) cos x C 4 4
Câu 82. Nguyên hàm F x h 2 f x
tan x hỏ ã F 0 3 là: A. f x tan x x 3 B. f x tan x x 3 C. f x tan x x 3 D. f x tan x x 3 13 Câu 83. N 2 f x cos xf 0 thì: 4 4 1 1 1 A. f x x cos 2x 3 B. f x x cos 2x 4 2 2 2 7 1 C. f x sin x D. f x cos 2x 3 2 2
Câu 84. Ng h h y f x sin x cosx 1 là: A. F x sinx cosx C B. F x sin x cosx x C C. F x cosx sin x x C D. F x sin x cosx x C Câu 85. K i g ? 1 A. sin x.cosxdx cosx.sin x C B. sin x.cos xdx cos 2x C 2 3 cos x 3 sin x C. 2 cos x.sin xdx C D. 2 sin x.cos xdx C 3 3 Câu 86. K i g ? 1 1 1 A. cos 3x.cos xdx sin 4x sin 2x C 2 4 2 1 1 1 B. sin 3x.cos xdx cos 4x cos 2x C 2 4 2 1 C. sin 3x.cos xdx cos 3x.sin x C 3 cos 2x D. sin x.cos xdx C 4 cos 2x
Câu 87. Ng h h y = dx là: 2 2 sin x.cos x A. F x cosx – sin x C B. F x cosx sin x C C. F x cotx – tan x C D. F x cotx – tan x C
Câu 88. Tìm nguyên hàm
2 sin 3x cos2x.dx ? 1 1 1 A. F x cos 5x cos x C B. F x cos 5x cos x C 5 3 2
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1 1 1 C. F x cos 5x cos x C D. F x cos 5x cos x C 2 3 5
Câu 89. Tìm nguyên hàm: 2 sin 2xdx 1 1 1 1 1 1 1 A. x sin 4x C B. 3 sin 2x C C. x sin 4x C D. x sin 4x C 2 8 3 2 8 2 4 1
Câu 90. Tìm nguyên hàm dx = 2 2 sin x.cos x A. 2 tan 2x C B. 2 cot2x C C. 4 cot2x C D. 2 cot2x C Câu 91. K i g ? x 1 x 1 A. 2 cos 2xdx sin 4x C B. 2 sin 2xdx sin 4x C 2 8 2 8 1 C. cos 4xdx sin 4x C D. 2 2 sin 2xdx cos 2x C 4 Câu 92. K i g ? A. 2 cot xdx cotx x C B. 2 tan xdx tan x x C 3 tan x 1 sin 2x C. 2 tan xdx C D. 2 sin xdx x C 3 2 2
Câu 93. T h f x i f ' x sin x cos x f 0 . 4 A. f x cosx sin x 2 ` B. f x cosx sin x 2 2 2 C. f x cos x sin x D. f x cos x sin x 2 2
NHÓM 4: HÀM SỐ MŨ, LOGARIT Câu 94. T g h h ( ) x x f x e e . A. x x x f x d e e C . B. x x x f x d e e C . C. x x x f x d e e C . D. x x x f x d e e C . Câu 95. T g h h x 2x f (x) 2 .3 . x x 2 1 9 1 A. f x x d . C . B. f x x d . C . 9 ln 2 ln 9 2 ln 2 ln 9 x x 2 1 2 1 C. f x x d . C . D. f x x d . C . 3 ln 2 ln 9 9 ln 2 ln 9 Câu 96. H g h h ( ) x (3 x f x e e ) là A. ( ) 3 x F x e x C . B. ( ) 3 x x ln x F x e e e C .
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1 C. F(x) 3 x e C . D. ( ) 3 x F x e x C . x e Câu 97. H 7 x F x e tan x g h h ? x e x 1 A. x f x e 7 . B. f x 7e . 2 cos x 2 cos x x 1 C. x 2 f x 7e tan x 1. D. f x 7 e . 2 cos x Câu 98. T g h h 4x 2 f (x) e . 1 A. 2x 1 f x x d e C . B. 2x 1 f x x d e C . 2 1 1 C. 4x 2 f x x d e C . D. 2x 1 f x x d e C . 2 2 Câu 99. Tính x e 4 dx 1 A. x e 4x C B. 4x C C. x e C D. x e 4x C x e x 1 Câu 100. Tính 3 1 e dx 2 x 1 1 1 1 x 1 x 1 A. 3 1 e C B. 3 1 3 x e C C. 3 1 3 x e C D. 3 1 e C 3 x x x 3 x Câu 101. M g h h x 2 f x 9 3x A. 3 9x F x x B. x 3 F x 9 ln 9 x 9x 9x C. 3 F x x D. 3 F x x 9 ln 9 x e
Câu 102. M g h F x h x f x e 1 hỏ F 1 e là 2 x x 1 x 1 A. F x e 1 B. F x e 1 x x x 1 x 1 C. F x e 1 D. F x e 1 x x Câu 103. H 1 ( ) x f x e ó g h A. 1 ( ) x F x e C B. 1 ( ) x F x e 1 C. 2 ( ) x F x e C D. 1 ( ) x F x e C 2 x
Câu 104. G i F(x) ậ h g h h 1 ( ) 3x f x thì F(x) là: A. 1 ( ) 3x F x C B. ( ) 3 ln 3.3x F x C
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN x 1 3 C. 1 ( ) 3 ln 3.3x F x C D. F(x) C ln 3
Câu 105. G i F(x) ậ h g h h ( ) x f x e
2 thì F(x) là: A. ( ) x F x e 2 C B. ( ) x F x e 2x C 1 C. x 1 F(x) e 2x C D. x 1 F(x) e 2x C x 1 Câu 106. Ng h h 2 ( ) x x f x e e là: 1 A. 2x x e e C B. 2 2 x x e e C C. x ( x e e x) C D. K t qu khác 2 x 1
Câu 107. Ng h h f (x) 2e là: 2 cos x x e A. 2 x e tanx C B. x e 2x C. x e tanx C D. K t qu khác 2 cos x Câu 108. Tính (3 cos 3x x )dx 3x 3x 3x A. 3 sin x
C B. 3 sin x C C. 3 sin x C ln 3 ln 3 ln 3 3x D. 3 sin x C ln 3 Câu 109. H x F x e tan x
C g h h f (x)nào? x 1 x 1 x 1 A. f (x) e B. f (x) e C. f (x) e D. K t qu khác 2 sin x 2 sin x 2 cos x Câu 110. N ( ) x f x dx e sin 2x
C thì f (x) g x 1 A. x e cos2x B. x e cos2x C. x e 2 cos2x D. e cos 2x 2 Câu 111. T g h h 2x 4x f x . 2 x 2x 2 2x A. F(x) C B. x 1 F(x) 1 2 C ln 2 ln 2 ln 2 2x 4x 2x C. F(x) 1 C D. ( ) 1 2x F x C . ln 2 ln 2 2 ln 2 x 1 Câu 112. Tìm 3 1 e dx . 2 x 1 1 x 1 A. 3 1 F(x) e C B. 3 1 ( ) 3 x F x e C 3 x x 1 1 x 1 C. 3 1 ( ) 3 x F x e C D. 3 1 F(x) e C x 3 x
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN x 2 Câu 113. Tìm e 3 . 5 x x e x 1 x 1 A. F(x) 3e C B. F(x) 3e C 4 2x 4 2x x 1 x 1 C. F(x) 3e C D. F(x) 3e C 4 2x 4 2x Câu 114. Ng h h 2x 3 y f x e 2x 1 là: 2x 3 e A. 2x 3 2 F x e x x C B. F x 2x C 2 2x 3 e C. 2 F x x x C D. 2 3 2 2 x F x e x x C 2 Câu 115. Ng h h 3x 2x 1 y f x 2 4 là 3x 2x 1 2 4 3x 2x 1 2 4 A F x C B.F x C 2.ln 8 4.ln16 ln 2 ln 4 3x 2x 1 2 4 3x 2x 1 3.2 2.4 C. F x C D.F x C ln 8 ln16 ln 2 ln 4 Câu 116. K i g ? x x x 2x A. x 2x x 2x e dx e C B. e 2 dx e C ln 2 3 x 2 e 1 1 1 C. x x e e 1 dx C D. xdx C 3 x x e e 2
Câu 117. Tìm nguyên hàm 3 2 x e dx 3x 6 4 x e e 3x 6 4 x e e A. 3x C B. 4x C 3 6 3 6 3x 6 4 5 x e e 3x 6 4 x e e C. 4x C D. 4x C 3 6 3 6
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
Dạng toán 2. TÍNH NGUYÊN HÀM CỦA HÀM SỐ HỮU TỶ ⓣⓗⓑⓣⓝ Phƣơng Pháp
ài toán tổng quát T h g h với v hứ hô g ă .
Phƣơng pháp giải: — N ậ ử ậ ẫ hi hứ . — N ậ ử ậ ẫ Xe é ẫ v hi ó
+ N ẫ h h h h h ẽ ử ụ g ồ g h hứ ể v ạ g ổ g h .
M ờ g h ồ g h hứ h ờ g gặ với
+ N ẫ hô g h h h h h i ổi v v ạ g g gi .
BÀI TẬP VẬN DỤNG BT 1. T h g h 2x 1 a) I dx ĐS: I 2x 3 ln x 1 C. x 1
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............................................................................................................................................................. 3x 1 b) I dx ĐS: I 3x 7 ln x 2 C. x 2
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3x 1 3 1 c) I dx ĐS: I C. 3 (x 1) 2 x 1 (x 1)
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............................................................................................................................................................. x 1 x 1 d) I dx ĐS: I ln 2x 3 C. 2x 3 2 4
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............................................................................................................................................................. 4 2 x 3x 2x 1 3 x 1 e) I dx ĐS: I 3x 2 ln x C. 2 x 3 x
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............................................................................................................................................................. 2 x x 1 2 x f) I dx ĐS: I x 3 ln x 2 C. x 2 2
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............................................................................................................................................................. 2 4x 6x 1 1 g) I dx ĐS: 2 I x 2x ln 2x 1 C. 2x 1 2
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............................................................................................................................................................. 3 2 4x 4x 1 3 2 2x x x 1 h) I dx ĐS: I ln 2x 1 C. 2x 1 3 2 2 2
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............................................................................................................................................................. dx x i) I ĐS: I ln C. x (x 1) x 1
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............................................................................................................................................................. dx 1 x 2 j) I ĐS: I ln C. 2 x 4 4 x 2
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............................................................................................................................................................. dx 1 k) I ĐS: I C. 2 x 6x 9 x 3
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26 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN dx 1 x 5 l) I ĐS: I ln C. 2 x 6x 5 4 x 1
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............................................................................................................................................................. dx 1 x 1 m) I ĐS: I ln C. 2 2x x 3 5 2x 3
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............................................................................................................................................................. dx 1 2x 5 n) I ĐS: I ln C. 2 2x 7x 5 3 x 1
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............................................................................................................................................................. 4x 5 o) I dx ĐS: I ln x 2 3 ln x 1 C. 2 x x 2
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............................................................................................................................................................. 4x 11 p) I dx ĐS: I 3 ln x 2 ln x 3 C. 2 x 5x 6
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............................................................................................................................................................. x 1 1 q) I dx ĐS: I ln x 2 4 ln x 3 C. 2 x x 6 5
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............................................................................................................................................................. 5x 3 r) I dx ĐS: I 2 ln x 1 7 ln x 2 C. 2 x 3x 2
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............................................................................................................................................................. 1 2x 1 3 s) I dx ĐS: I ln x ln x 2 C. 2 x 2x 2 2
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............................................................................................................................................................. 2 x dx t) I ĐS: I x 16 ln x 4 9 ln x 3 C. 2 x 7x 12
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
............................................................................................................................................................. 2 x 1 x 1 u) I dx ĐS: I x ln C. 2 x 1 x 1
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............................................................................................................................................................. 2 x 1 v) I dx ĐS: 2 I x 2x 4 ln x 2 C. x 2 2
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............................................................................................................................................................. 3x 2 3 7 w) I dx ĐS: I ln 2x 1 C. 2 4x 4x 1 4 4(2x 1)
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............................................................................................................................................................. 2 4 3x x 3 x 2 x 2 x) I dx ĐS: I x ln C. 2 x 2 3 2 x 2
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............................................................................................................................................................. 2 x x 2 y) I dx ĐS: I x 3 ln x 2 C. 2 (x 2) x 2
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............................................................................................................................................................. 2 x .dx 1 x 1 1 1 z) I ĐS: I ln C. 2 2 (1 x ) 4 x 1 x 1 x 1
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............................................................................................................................................................. BT 2. T h g h 2 2x 5x 3 3 5 a) I dx ĐS: I ln x 2 ln x 1 ln x 2 C. 3 2 x x 2x 2 2
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28 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 2x 8x 10 1 20 17 b) I dx ĐS: I ln x 2 ln x 1 ln x 2 C. 3 2 x x 4x 4 6 3 2
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............................................................................................................................................................. 3 x 1 1 9 28 c) I dx ĐS: I x ln x ln x 2 ln x 3 C. 3 2 x 5x 6x 6 2 3
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............................................................................................................................................................. 2 3x 3x 3 3 d) I dx ĐS: I 2 ln x 1 ln x 2 C. 3 x 3x 2 x 1
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............................................................................................................................................................. dx 1 e) I ĐS: 3 I ln x ln x 1 C. 3 x (x 1) 3
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BÀI TẬP TRẮC NGHIỆM 3x 5
Câu 1. M g ên hà hà y là: x 2 A. F(x) 3x 4 ln x 2 C B. F(x) 3x ln x 2 C C. F(x) 3x ln x 2 C D. F(x) 3x ln x 2 C x
Câu 2. M g h h f (x) là: x 1 A. ln x 1 B. x ln x 1 C. x ln x 1 D. 2 ln x 1
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 x 2x 1
Câu 3. h h f (x)
. M g h F(x) f (x) hỏ F(1) 0 là: 2 x 2x 1 2 2 2 2 A. x 2 B. x 2 C. x 2 ln x 1 D. x 2 x 1 x 1 x 1 x 2 x
Câu 4. H hô g g h h f x ? 2 x 1 2 x x 1 2 x x 1 2 x 2 x x 1 A. B. C. D. x 1 x 1 x 1 x 1 2 2 x 1 Câu 5. h h f x
. M g h F x f x hỏ F 1 4 là : 3 x 2 x 2 2 x 1 A. 2 ln x 4 B. 2 ln x 4 2 2 x 2 2 2x 2 x 2 C. 2 ln x 4 D. 3 F x x 2x C 2 2 x 3 x 1
Câu 6. Ng h h f x là: x 1 3 2 x x 3 2 x x A. F x x 2 ln x 1 C B. F x x 2 ln x 1 C 3 2 3 2 3 2 x x 3 2 x x C. F x x ln x 1 C D. F x x 2 ln x 1 C 3 2 3 2 3 2 x 3x 3x 1 1
Câu 7. G i h F(x) g h f (x) i F(1) . Vậ 2 x 2x 1 3 F(x) là: 2 x 2 13 2 x 2 13 A. F(x) x B. F(x) x 2 x 1 6 2 x 1 6 2 x 1 2 x 2 C. F(x) x C D. F(x) x 2 x 1 2 x 1 2 x 2x 1 1
Câu 8. T g h F h f (x) i F(1) . K x 2 2 x 2 x A. F(x) 2x ln x 2 B. F(x) 2x ln x 2 2 2 2 x 1 2 x 1 C. F(x) 2x ln x D. F(x) 2x ln x 2 2 2 2 A 3 2 3x 3x 3 A B C
Câu 9. Ta có: f (x) B 2 . 3 2 x 3x 2 x 1 x 2 x 1 C 1 Tính f (x)dx F(x) C
30 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3 2 1 A. F(x) C 2 x 1 x 2 x 1 3 B. F(x) 2 ln x 1 ln x 2 C x 1 2 C. F(x) 3 ln x 1 ln x 2 C x 1 1 D. F(x) 3 ln x 1 2 ln x 2 C x 1 1 1
Câu 10. Ng h h f (x) là : 2 x x 1 1 A. 2 lnx lnx C B. ln x C C. ln x C D. K t qu khác x x 1
Câu 11. Tính nguyên hàm dx 2x 1 1 1 A. ln 2x 1 C B. ln 2x 1 C C. ln 2x 1
C D. ln 2x 1 C 2 2 4 2x 3
Câu 12. Ng h h f x = là : 2 x 3 2x 3 3 2x 3 3 2x A. C B. C C. 2 3 ln x
C D. K t qu khác 3 x 2 3 x 3 x Câu 13. K dx là: 2 1 x 1 1 A. 2 1 x C B. C C. C D. 2 1 x C 2 1 x 2 1 x x
Câu 14. M g h h f x là: x 1 A. F(x) ln x 1 B. F(x) x ln x 1 C. F(x) x ln x 1 D. F(x) 2 ln x 1 Câu 15. H ới g h h 4 4 f x sin x cos x ? 1 1 A. F(x) x sin 4x B. F(x) x sin 4x 4 4 3 1 3 1 C. F(x) x sin 4x D. F(x) x cos 4x 4 16 4 4 1
Câu 16. M g h F(x h f x là 2x 5 1 A. F(x) ln 2x 5 2016 B. F(x) ln 2x 5 2
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 1 C. F(x) D. F(x) 2 2 2x 5 2x 5 1
Câu 17. Ng h h y f x là: 2 1 2x 1 1 2 A. F x . C B. F x ln 1 2x C 2 1 2x 1 1 1 C. F x . C D. F x C 2 1 2x 1 2x 1
Câu 18. Ng h h y f x sin 2x là: 4x 3 1 1 1 1 A. F x .cos 2x .ln 4x 3 C B. F x .cos 2x .ln 4x 3 C 2 4 2 4 1 1 C. F x cos 2x .ln 4x 3 C D. F x .cos 2x ln 4x 3 C 4 2 Câu 19. K i g ? 4 2x 1 A. dx 4 ln 2x 1 C B. dx 2x ln x C 2x 1 x 2 2x 1 C. dx ln 2x 1 C D. dx 2x ln x 1 C 2x 1 x 1 Câu 20. K i g ? 2x 1 x 5 x 1 A. dx ln 3 4x C B. dx x 2 ln x 1 C 3 4x 2 8 x 1 2 6 C. dx 2 ln x 1 C D. dx 3 ln 2x 3 C x 1 2x 3 3 x
Câu 21. Ng h h y dx là: x 1 1 1 1 1 A. 3 2 x x x ln x 1 C B. 3 2 x x x ln x 1 C 3 2 3 2 1 1 1 1 C. 3 2 x x x ln x 1 C D. 3 2 x x x ln x 1 C 6 2 3 4 2x 3
Câu 22. Ng h h I dx. 2 là: 2x x 1 2 5 2 5 A. F(x) = ln 2x 1 ln x 1 C B. F(x) = ln 2x 1 ln x 1 C 3 3 5 2 2 5 2 5 C. F(x) = ln 2x 1 ln x 1 C D. F(x) = ln 2x 1 ln x 1 C 3 3 3 3
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
Dạng toán 3. TÍNH NGUYÊN HÀM ẰNG PHƢƠNG PHÁP ĐỔI IẾN SỐ ⓣⓗⓑⓣⓝ
A – PHƢƠNG PHÁP GIẢI TOÁN Đ nh : Cho f (u)du
F(u) C v u (
u x) h ó ạ h i ụ h f (
u x) u (x) dx F ( u x) C.
1. Đổi i n ạng 1t (x). I f (ax b)n xdx PP t ax b dt a.dx m n x PP n 1 I dx t x 1 dt (n 1) n x .dx , với , m n . n 1 ax 1 2 n PP 2 I f (ax b) xdx t ax b dt 2ax.dx n I
f (x) f (x) dx PP Đặ n t
f (x), ừ ờ g h ổi i ạ g 2. 1 I f (ln x) dx t ln x x PP Đặ 1 t a b ln x I f (a b ln x) dx x ( x ) x I f e e dx PP Đặ x t e . I
f (cosx) sin xdx PP Đặ t cosx dt sinxdx. I
f (sin x) cosxdx PP Đặ t sinx dt cosxdx. 1 1 I f (tan x) dx PP Đặ 2 t tan x dt dx (1 tan x)dx. 2 cos x 2 cos x 1 1 I f (cotx) dx PP Đặ 2 t cotx dt dx (1 cot x)dx. 2 sin x 2 sin x 2 t sin x dt sin 2xdx 2 2 I
f (sin x;cos x) sin 2xdx PP Đặ 2 t cos x dt sin 2xdx I f (sin x cos x) (sin x cos x) dx PP Đặ t sinx cosx.
2. Đổi i n ạng 2x (t). 2 2 2 ( ) n I f a x x dx PP Đặ x . a sint dx .
a cost.dt. adt 2 2 2 ( ) n I f x a x dx PP Đặ x a.tant dx 2 cos t a a sint 2 2 2 ( ) n I f x a x dx PP Đặ x dx dt 2 cost cos t
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN dx 1 dt I PP Đặ x a dx n 2 (x a) . ax bx c 2 t t n n 1 I R ax b,..., k ax b dx PP Đặ n t ax b với n .
BC.N.N n ;n ;...;n 1 2 k x a 0 t x a x b khi dx x b 0 I PP Đặ (x a)(x b) x a 0 t x a x b khi x b 0
B - BÀI TẬP VẬN DỤNG BT 3. T h g h 2016 2017 (1 x) (1 x) a) 2015 I x (1 x) dx ĐS: I C. 2016 2017
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............................................................................................................................................................. 4 2 (x 1) b) 2 3 I x (x 1) dx ĐS: I C. 8
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............................................................................................................................................................. 12 11 10 (x 1) 2(x 1) (x 1) c) 2 9 I x (x 1) dx ĐS: I C. 12 11 10
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............................................................................................................................................................. 2 10 2 9 (2 3x ) (2 3x ) d) 3 2 8 I x (2 3x ) dx ĐS: I C. 180 81
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34 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN xdx 1 e) I ĐS: 2 I ln x 2 C. 2 x 2 2
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............................................................................................................................................................. 2x 2 f) I dx ĐS: I 2 ln x 1 C. 2 (x 1) x 1
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............................................................................................................................................................. x 1 1 1 1 g) I dx ĐS: I C. 5 (x 1) 3 (x 1) 4 x 1 3
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............................................................................................................................................................. 3 x 1 1 h) I dx ĐS: I C. 2 1 x 2 2 2 2(1 x ) 4(1 x )
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............................................................................................................................................................. 3 1 1 1 1 3 1 i) I 1 dx ĐS: I C. 2 x x 4 3 2 4x x 2x x
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............................................................................................................................................................. xdx 1 j) I ĐS: I ln x 1 C. 2 (x 1) x 1
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
............................................................................................................................................................. xdx 1 1 1 k) I ĐS: I C. 3 (2x 1) 2 2 4(2x 1) 2(2x 1)
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............................................................................................................................................................. 5 x 4 2 x x 1 l) I dx ĐS: 2 I ln(x 1) C. 2 x 1 4 2 2
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............................................................................................................................................................. 4 x 5 1 x 2 m) I dx ĐS: I ln C. 10 x 4 5 20 x 2
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............................................................................................................................................................. 3 x dx 3 2 x x n) I ĐS: I x ln x 1 C. x 1 3 2
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............................................................................................................................................................. BT 4. T h g h (x 1)dx a) I ĐS: 2 I x 2x 4 C. 2 x 2x 4
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............................................................................................................................................................. 2 3 (2 x ) b) 2 I
x. 2 x .dx. ĐS: I C. 3
36 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. 2xdx 3 c) I ĐS: 3 2 2 I (x 4) C. 3 2 x 4 2
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............................................................................................................................................................. 2 x dx 2 2(3x 4x 8) 1 x d) I ĐS: I C. 1 x 15
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............................................................................................................................................................. 4 15 e) 3 2 I
5x. 1 x .dx. ĐS: 2 3 I (1 x ) C. 8
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............................................................................................................................................................. 4 2 3 2 2 3(20x 4x 3). (1 2x ) f) 5 3 2 2 I
x . (1 2x ) .dx. ĐS: I C. 320
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............................................................................................................................................................. 4x 1 g) I dx. ĐS: I 2x 1 4 2x 1 5 ln 2x 1 2 . C 2x 1 2
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3 x 2 3 (4 x ) h) I dx. ĐS: 2 I 4 4 x C. 2 4 x 3
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............................................................................................................................................................. dx 2 1 x 4 2 i) I ĐS: I ln C. 2 x x 4 2 4 x 4 2
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............................................................................................................................................................. 3 2 2x 3x x 2 3 2 (x x 1) j) I dx. ĐS: 2 I 2 x x 1 C. 2 x x 1 3
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............................................................................................................................................................. 2 k) 3 I
sin x. cos x .dx. ĐS: 3 I (3 cos x 7 cos ) x cos x . C 21
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............................................................................................................................................................. dx 1 x e 1 l) I ĐS: I ln C. 1 x e 1 x e 1
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............................................................................................................................................................. dx 2 1 1 3 ln x 1 m) I ĐS: I ln C. 2 x ln x. 1 3 ln x 2 2 1 3 ln x 1
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38 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
.............................................................................................................................................................
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............................................................................................................................................................. xdx 3 2 3 x (x 1) n) I ĐS: I C. 2 x x 1 3 3
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............................................................................................................................................................. BT 5. T h g h 1 3 ln x a) 2 I ln x dx ĐS: I C. x 3
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............................................................................................................................................................. 3 ln x 1 b) I dx ĐS: I 3 ln x ln ln x C. x ln x
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............................................................................................................................................................. 1 2 (1 ln x) c) I (1 ln x) dx ĐS: I C. x 2
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............................................................................................................................................................. ln x 1 3 2 (1 ln x) d) I dx ĐS: I 2 1 ln x C. 1 ln x x 3
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3 2 ln x. 2 ln xdx 3 e) I ĐS: 3 2 4 I (2 ln x) C. x 8
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............................................................................................................................................................. 3 log x 2 3 1 (1 3 ln x) f) 2 I dx ĐS: 2 I 1 3 ln x C. 2 x 1 3 ln x 3 9 ln 2 3
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............................................................................................................................................................. BT 6. T h g h dx x e 1 a) I ĐS: I ln C. x e 1 x e
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............................................................................................................................................................. dx 1 x e b) I ĐS: I ln C. x e 2 2 x e 2
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............................................................................................................................................................. dx x e 2 c) I ĐS: I ln C. x e 2 x e 3 x e 1
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............................................................................................................................................................. x e 1 d) I dx ĐS: 2 ln x I e 1 C. x x e e 2
40 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
.............................................................................................................................................................
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............................................................................................................................................................. dx 1 x e 2 e) I ĐS: I ln C. x e 4. x e 4 x e 2
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............................................................................................................................................................. x 3 (1 e ) x 2 (1 e ) x 1 f) I dx ĐS: I 2(1 e ) 3x C. x e 2 x e
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............................................................................................................................................................. 2x e 3 x e 1 3 x e x x 1 g) I dx ĐS: 2 I ln(e 3e 2) ln C. 2x e 3 x e 2 2 2 x e 2
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............................................................................................................................................................. 2x e x 3 2 (e 1) h) I dx ĐS: I 2 x e 1 C. x e 1 3
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............................................................................................................................................................. dx x e 1 2 i) I ĐS: I 2 ln C. x x e e x x e e
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ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC – TRUNG – NAM 41 | THBTN
TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN dx x e j) I ĐS: I 2 ln C. x e 1 x e 1
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............................................................................................................................................................. BT 7. T h g h cos xdx a) I ĐS: I ln 1 sin x C. 1 sin x
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............................................................................................................................................................. (2 sin x 3)cos x 1 b) I dx ĐS: I (2 sin x 1) 2 ln 2 sin x 1 . C 2 sin x 1 2
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............................................................................................................................................................. 3 cos xdx 3 c) I ĐS: I C. 2 (1 sin x) 1 sin x
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............................................................................................................................................................. 2 cos xdx 3 d) I ĐS: I ln sin x C. 3 2 sin x 2
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............................................................................................................................................................. 3 5 sin x sin x e) 2 3 I
sin x.cos x.dx. ĐS: I C. 3 5
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42 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
.............................................................................................................................................................
............................................................................................................................................................. 2 1 2 sin x 1 f) I dx ĐS: I ln 1 sin2x C. 1 sin 2x 2
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............................................................................................................................................................. 3 cos x 1 g) I dx ĐS: I sin x C. 2 sin x sin x
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............................................................................................................................................................. sin 2x 4 h) I dx ĐS: I 2 ln(2 sin ) x C. 2 (2 sin x) 2 sin x
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............................................................................................................................................................. i) sin x I e .cosx.dx. ĐS: sin x I e C.
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............................................................................................................................................................. 5 3 sin x 2 sin x x sin 2x j) 3 2 I (cos x 1).cos x.dx. ĐS: I sin x C. 5 3 2 4
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............................................................................................................................................................. cos xdx 3 sin x k) I ĐS: I ln C. 2 6 5 sin x sin x 2 sin x
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. cos xdx 1 5 sin x l) I ĐS: I ln C. 2 11 7 sin x cos x 3 2 sin x
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............................................................................................................................................................. BT 8. T h g h sin xdx a) I ĐS: I ln 1 cos x C. 1 cos x
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............................................................................................................................................................. 3 4 sin x b) I dx ĐS: 2 I 4 cos x cos2x C. 1 cos x
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............................................................................................................................................................. 5 3 cos x cos x c) 2 3 I cos x sin xdx ĐS: I C. 5 3
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............................................................................................................................................................. sin 2x cos x d) I dx ĐS: 2 I 2 cos x cos x ln cos x 1 . C 1 cos x
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............................................................................................................................................................. sin 4x 3 cos2x e) I dx ĐS: I 6 ln( ) 2 cos2x 6 . C 2 1 cos x 2
44 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. sin x sin 3x 2 2 cos x 1 f) I dx ĐS: I ln 2 cos x C. cos 2x 2 2 cos x 1
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............................................................................................................................................................. x tan .sin x.(1 sin x) 4 2 1 g) I dx ĐS: I C. 3 cos x cos x
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............................................................................................................................................................. 3 sin x 1 1 h) I dx ĐS: I C. 4 cos x 3 3 cos x cos x
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............................................................................................................................................................. BT 9. T h g h tan x 2 tan x a) I dx ĐS: I C. 2 cos x 2
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............................................................................................................................................................. 4 sin x 5 tan x b) I dx ĐS: I C. 6 cos x 5
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ADMIN TRẦN VĂN TÀI – 0977.413.341 - TOÁN HỌC BẮC – TRUNG – NAM 45 | THBTN
TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
............................................................................................................................................................. 4 tan x 3 tan x 1 tan x 1 c) I dx ĐS: I tan x ln C. cos 2x 3 2 tan x 1
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............................................................................................................................................................. dx 1 3 tan x 5 d) I ĐS: I ln C. 2 2 5 cos x 8 sin x cos x 3 sin x 2 tan x 1
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............................................................................................................................................................. (1 sin 2x) dx 2 tan x 3 tan x 1 e) I ĐS: I ln 2 tan x 1 C. 3 4 2 sin x cos x cos x 4 4 8
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............................................................................................................................................................. dx 3 tan x 1 f) I ĐS: I 2 tan x C. 4 2 cos x sin x 3 tan x
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............................................................................................................................................................. dx g) I ĐS: I 2 ln 1 tan x C. cos x cos x 4
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............................................................................................................................................................. tan x 4 1 h) I dx ĐS: I C. cos 2x 1 tan x
46 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. BT 10. T h g h 2 cos x 1 a) I dx ĐS: 3 I cot x C. 4 sin x 3
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............................................................................................................................................................. 2 cos x 7 5 3 15 cot x 42 cot x 35 cot x b) I dx ĐS: I C. 8 sin x 105
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............................................................................................................................................................. dx 4 c) I ĐS: 4 3 I cot x C. 2 4 sin x cotx 3
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............................................................................................................................................................. dx 1 d) I ĐS: 2 I ln cot x cot x C. 3 cos x sin x 2
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............................................................................................................................................................. dx e) I ĐS: I 2 ln cot x C. sin x sin x 6
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
............................................................................................................................................................. sin x.dx 1 f) I ĐS: I C. 3 (sin x cos x) 2 2(1 cot x)
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............................................................................................................................................................. BT 11. T h g h sin x cos x a) I dx ĐS: I ln sin x cos x C. sin x cos x
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............................................................................................................................................................. cos 2xdx b) I ĐS: I sin x cos x 2 2 ln sin x cos x 2 sin x cos x 2
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............................................................................................................................................................. cos 2x dx 1 1 c) I ĐS: I C. 3 (sin x cos x 2) 2 (sin x cos x 2) sin x cos x 2
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............................................................................................................................................................. 2 4 (1 sin x) d) 2 3 I sin 2x.(1 sin x) .dx. ĐS: I C. 4
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............................................................................................................................................................. BT 12. T h g h dx a) I ĐS: I arcsin x C. 2 1 x
48 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. dx 2 9 x b) I ĐS: I C. 2 2 x 9 x 9x
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............................................................................................................................................................. dx x c) I ĐS: I C. 2 3 (x 1) 2 x 1
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............................................................................................................................................................. dx 2 25 x d) I ĐS: I C. 2 2 x 25 x 25x
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............................................................................................................................................................. 3 x 1 e) I dx ĐS: 2 2 I (x 2) 1 x C. 2 1 x 3
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............................................................................................................................................................. 3 2 1 x 2 2 (1 x ) f) I dx ĐS: I C. 4 x 3 3x
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN dx g) I ĐS: 2 I ln x x 4 C. 2 x 4
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C - BÀI TẬP TRẮC NGHIỆM x 1 Câu 118. Tính dx 2 x 2x 5 2x 2 A. C B. 2 2 x 2x 5 C 2 x 2x 5 2 x 2x 5 C. C D. 2 x 2x 5 C 2 x
Câu 119. H g h h f x là 2 x 1 A. 2 F x ln x 1 C B. 2 F x x 1 C 2 C. 2 F x 2 x 1 C D. F x C 2 3 x 1 Câu 120. M g h h sin cos . x f x x eA. sin x F x e B. cosx F x e C. sin x F x e D. sin sin . x F x x e 2016 Câu 121. h h 2 f x x x 1 . Khi ó 2017 2016 2 x 1 2 x 1 A. f x dx C B. f x dx 4034 4032 2016 2017 2 x 1 2 x 1 C. f x dx D. f x dx 2016 2017 2 Câu 122. H x F x e g h h 2 x 2 e 2 A. 2 x f x xe B. 2x f x e C. f x D. 2 x f x x e 1 2x
Câu 123. H g h h f (x) tanx là: 1 1 A. F(x) C B. F(x) C 2 cos x 2 sin x C. F(x) ln cosx C D. F(x) ln cos x C
50 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN Câu 124. K cosx s inx 1dx g 3 2 3 2 A. F(x) s in x 1 C B. F(x) s in x 1 C 3 3 2 3 2 C. F(x) s in x 1 C D. F(x) s in x 1 C 3 3 x e Câu 125. K dx g x e 3 A. ( ) x F x e 3 C B. ( ) 2 x F x e 3 C x e C. ( ) x F x e 3 C D. F(x) C x e 3x ln x
Câu 126. H f (x) có các nguyên hàm là: x 1 A. 2 F(x) ln x C B. F(x) ln x C 2 1 1 C. 2 F(x) ln x C D. F(x) C 2 2 x.x 1 x
Câu 127. H f (x) ln x( ) có các nguyên hàm là: x ln x 2 2 ln x x A. 2 2 F(x) ln x x C B. F(x) C 2 2 ln x 2 x C. 2 F(x) x C D. F(x) ln x(ln x ) C 2 2 ln x 1
Câu 128. G i h F(x) g h f (x) i F
1. Vậ F(x) là: s inx 2 1 1 cos x 1 1 cos x A. F(x) ln 1 B. F(x) ln 2 1 cos x 2 1 cos x 1 cos x 1 1 cos x C. F(x) ln 1 D. F(x) ln 1 1 cos x 2 1 cos x x
Câu 129. G i F(x) g h f (x) hỏ ã F(2) 0 . Khi ó h g 2 8 x trình F(x) x ó ghiệ A. x 0 B. x 1 C. x 1 D. x 1 3
Câu 130. Ng h h y = sin2x.cos3x là: 1 1 1 1 A. 3 5 sin x sin x C B. 3 5 sin x sin x C 3 5 3 5
C. sin3x sin5x + C D. Đ h .
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3 x
Câu 131. M g h h y là: 2 2 x 1 A. 2 F(x) x 2 x B. 2 2 x 4 2 x 3 1 1 C. 2 2 x 2 x D. 2 2 x 4 2 x 3 3 1
Câu 132. H ới g h h y 2 4 x A. 2 F(x) ln x 4 x B. 2 F(x) ln x 4 x C. 2 F(x) 2 4 x D. 2 F(x) x 2 4 x Câu 133. M g h h 2 f (x) x sin 1 x là: A. 2 2 2 F(x) 1 x cos 1 x sin 1 x B. 2 2 2 F(x) 1 x cos 1 x sin 1 x C. 2 2 2 F(x) 1 x cos 1 x sin 1 x D. 2 2 2 F(x) 1 x cos 1 x sin 1 x Câu 134. M g h h 2 f (x) x 1 x là: 2 1 3 1 A. 2 F(x) 1 x B. 2 F(x) 1 x 2 3 2 2 x 2 1 C. 2 F(x) 1 x D. 2 F(x) 1 x 2 3 2015
Câu 135. T g h F h f x x 1 x 1 2017 1 2016 A. F(x) 1 x 1 x C 2017 2016 1 2017 1 2016 B. F(x) 1 x 1 x C 2017 2016 1 2017 1 2016 C. F(x) 1 x 1 x C 2017 2016 1 2017 1 2016 D. F(x) 1 x 1 x C 2017 2016 2x
Câu 136. Tìm nguyên hàm F x i f (x) . K 2 x x 1 2 2 2 2 A. 3 2 2 F(x) x x 1 x 1 B. 3 2 2 F(x) x x 1 x 1 3 3 3 3 2 2 2 2 C. 3 2 2 F(x) x x 1 x 1 D. 3 2 2 F(x) x x 1 x 1 3 3 3 3
52 | THBT – CA GIÁO VIÊN CẦN FILE WORD LIÊN HỆ THẦY TÀI: 0977.413.341
TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 2 x 1
Câu 137. Tính: P dx x A. 2 P x x 1 x C B. 2 2 P x 1 ln x x 1 C 2 1 x 1 C. 2 P x 1 ln C D. Đ án khác. x sin x
Câu 138. Tìm nguyên hàm F x i f (x) . K sin x cos x 1 1 A. F(x) x ln sin x cos x C B. F(x) x ln sin x cos x C 2 2 1 1 C. F(x) x ln sin x cos x C D. F(x) x ln sin x cos x C 2 2
Câu 139. Tìm nguyên hàm F x i f x
cosx cos2x sin 4x . K 1 1 1 1 A. F(x) cos 7x cos 5x cos 3x cos x C 28 20 12 4 1 1 1 1 B. F(x) cos 7x cos 5x cos 3x cos x C 28 20 12 4 1 1 1 1 C. F(x) cos 7x cos 5x cos 3x cos x C 28 20 12 4 1 1 1 1 D. F(x) cos 7x cos 5x cos 3x cos x C 28 20 12 4
Câu 140. Tìm nguyên hàm F x i f (x) x sin x . K A. F(x) 2x cos x 4 x sin x 4 cos x C B. F(x) 2x cos x 4 x sin x 4 cos x C C. F(x) 2x cos x 4 x sin x 4 cos x C D. F(x) 2x cos x 4 x sin x 4 cos x C 2
Câu 141. Tính nguyên hàm x 1 xe dx 1 2 1 2 A. x 1 F(x) e C B. x 1 F(x) e C 2 2 1 2 1 2 C. x 1 F(x) e C D. ( ) x F x e C 2 2 ln 2 Câu 142. Tính 2 x dx . K i x A. ( ) 2 2 x F x 1 C B. ( ) 2 2 x F x 1 C C. ( ) 2 x F x C D. 1 ( ) 2 x F x C
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 1
Câu 143. H ới g h f (x) ? 2 1 x x A. F(x) B. 2 F(x) ln 1 x 2 1 x C. 2 F(x) ln x 1 x D. 2 F(x) ln x 1 x x
Câu 144. Ng h h f (x) là 2 x 1 1 A. 2 F(x) ln x 1 C B. 2 F(x) ln x 1 C 2 1 x 1 C. F(x) 2x C D. F(x) ln C 2 x 1 cos x Câu 145. Tìm dx . 20 sin x 1 1 A. F(x) C B. F(x) C 19 19 sin x 19 19 sin x 1 1 C. F(x) C D. F(x) C 19 19 cos x 19 19 cos x
Câu 146. H F(x) ln sin x 3 cosx g h h ? cos x 3 sin x cosx 3 sin x A. f (x) B. f (x) sin x 3 cos x sin x 3 cos x sin x 3 cos x C. f (x) cosx 3 sinx D. f (x) cos x 3 sin x x 1 Câu 147. Tìm dx . 2 x 2x 5 2x 2 A. C B. 2 2 x 2x 5 C 2 x 2x 5 2 x 2x 5 C. C D. 2 x 2x 5 C 2
Câu 148. Nguyên hàm F x h 2 3 f (x)
sin 2x.cos 2x hỏ F 0 là 2 1 1 1 1 A. 3 5 F x sin 2x sin 2x B. 3 5 F x sin 2x sin 2x 6 10 6 10 1 1 4 1 1 1 C. 3 5 F x sin 2x sin 2x D. 3 5 F x sin 2x sin 2x 6 10 15 6 10 15 x e
Câu 149. M g h F(x) h f (x) hỏ F 0 ln 3 là x e 2
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN A. ( ) ln x F x e 2 ln 3 B. ( ) ln x F x e 2 ln 3 C. ( ) ln x F x e 2 2 ln 3 D. ( ) ln x F x e 2 2 ln 3 2 x
Câu 150. Ng h h y f x là 3 x 1 3 3 x x 1 3 x 1 A. F x C B. F x C 3 3 3 2 x 1 3 2 x 1 C. F x C D. F x C 3 3
Câu 151. 0034: K i g ? 4 2 2 1 x 1 x 2 3 1 x A. 2 x 1 x dx C B. 2 2x 1 x dx C 3 4 4 2 2 3 1 x ln 1 x x C. 2 2 2x 1 x dx x . C D. dx C 4 2 1 x 2 Câu 152. K i g ? ln x A. 2 dx ln x C x 2 ln x ln x B. dx C x 2 5 3 3 2 ln x 3 2 ln ln 3 2 ln x x x C. dx C x 10 2 2 3 ln x ln x D. xdx C x 3 Câu 153. Ng h h 2x 3 2 x y f x e x là 2x 3x 2 e 2 x 2x 3x 2 e 2 x A. F x C B. F x C 2 ln 2 2 2 ln 8 2 3x 2 x 2x 3 2 x e x 2 C. 2 F x e C D. F x 1 C ln 8 2 2 ln 8 Câu 154. H F x ln sin x 3 cosx g h h g h cos x 3 sin x A. f x B. f x cosx 3 sin x sin x 3 cos x cos x 3 sin x sin x 3 cos x C. f x D. f x sin x 3 cos x cos x 3 sin x x ln 2 Câu 155. Tính 2 . dx i x
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN A. 2 2 x 1 C B. 2 2 x 1 C C. 1 2 x C D. 2 x C Câu 156. K i g dx 2 dx 1 x 1 1 A. ln ln ln x C B. ln C
x ln x.ln ln x 2 2 2 x x 1 x 1 1 dx 1 x xdx 1 C. tan C D. 2 ln 3 2x C 1 cos x 2 2 2 3 2x 4 Câu 157. T g h h 3 cos ( ) x f x e .sinx 1 A. 3 cos ( ) x f x dx e .cos x C B. 3 cos ( ) 3 x f x dx e C 3 1 C. 3 cos ( ) x f x dx e C D. 3 cos ( ) 3 x f x dx e .cosx C 3 ln x
Câu 158. T g h h f (x) 2x 2 ln x 2 ln x A. f (x)dx C B. f (x)dx C 4 2 2 ln x 1 C. f (x)dx C D. f (x)dx C 4x 2 2x Câu 159. Ng h h 3 I x x 1dx. là: 2 4 5 3 6 2 2 A. F(x) = x 1 x 1 x 1 x 1 x 1 C 9 7 5 3 2 4 6 3 6 2 2 B. F(x) = x 1 x 1 x 1 x 1 x 1 C 9 7 5 3 2 4 6 3 6 2 2 C. F(x) = x 1 x 1 x 1 x 1 x 1 C 9 7 7 3 2 4 6 3 6 2 1 D. F(x) = x 1 x 1 x 1 x 1 x 1 C 9 7 5 3 x d
Câu 160. Ng h h I là: 2x 1 4 A. F(x) = 2x 1 4 ln 2x 1 4
C B. F(x) = 2x 1 4 ln 2x 1 4 C 7 C. F(x) = 2x 1 4 ln 2x 1 4
C D. F(x) = 2x 1 ln 2x 1 4 C 2 5 cos x
Câu 161. Ng h h y dx là: 1 sin x 3 4 sin x cos x 3 4 sin 3x cos 4x A. cos x C B. sin x C 3 4 3 4
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN 3 4 sin x cos x 3 4 sin x cos x C. sin x C D. sin x C 3 4 9 4 2 (x x) x e
Câu 162. Ng h h y dx là: x x e A. F(x) = x 1 ln x xe xe 1 C B. F(x) = x 1 ln x e xe 1 C C. F(x) = x 1 ln x xe xe 1 C D. F(x) = x 1 ln x xe xe 1 C dx
Câu 163. Ng h h y là: 2 2 x a 1 x a 1 x a 1 x a 1 x a A. ln +C B. ln +C C. ln +C D. ln +C 2a x a 2a x a a x a a x a dx
Câu 164. Ng h h y là: 2 2 a x 1 a x 1 a x 1 x a 1 x a A. ln +C B. ln +C C. ln +C D. ln +C 2a a x 2a a x a x a a x a
Câu 165. Ng h h y x 4x 7 dx là: 5 3 1 2 2 5 3 1 2 2 A. 2 2 4x 7 7 4x 7 C B. 2 2 4x 7 7 4x 7 C 20 5 3 18 5 3 5 3 1 2 2 5 3 1 2 2 C. 2 2 4x 7 7 4x 7 C D. 2 2 4x 7 7 4x 7 C 14 5 3 16 5 3
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
Dạng toán 4. TÍNH NGUYÊN HÀM ẰNG PHƢƠNG PHÁP NGUYÊN HÀM TỪNG PHẦN ⓣⓗⓑⓣⓝ Phƣơng Pháp Đ nh N h i h v ó ạ h v i ụ h hay
Vận ụng giải toán:
Nhận ạng T h 2 h h ại h h hẳ g hạ ũ h g gi — Đ t: Suy ra:
Thứ tự ƣu tiên ch n u: l – – lượ – ũ v
p ầ ò lạ . Nghĩ u có ln hay thì ch n hay và còn lại. N u không có thì ch n hức và
còn lại. N hô g ó g hứ h g gi ….
Lƣu g ậ hứ v ậ g ứ g với ầ g h .
Dạng mũ nhân ƣợng giác ạ g g h ừ g hầ hồi.
BÀI TẬP VẬN DỤNG BT 13. T h g h a) I x sin x dx ĐS: I sinx cos x C.
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............................................................................................................................................................. b) (1 2 ) x I x e dx ĐS: (3 2 ) x I x e C.
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............................................................................................................................................................. x e c) x I e cosx dx ĐS: I (sin x cos x) C. 2
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. 2 x d) I (2x 1) ln x dx ĐS: 2 I (x x)ln x x C. 2
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............................................................................................................................................................. 3x 3x xe e e) 3x I x e dx ĐS: I C. 3 9
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............................................................................................................................................................. 3 3 x ln 2x x f) 2 I x ln 2x dx ĐS: I C. 3 9
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............................................................................................................................................................. g) I ln x dx ĐS: I x lnx x C.
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............................................................................................................................................................. x 1 1 h) I (x 1) sin 2x dx ĐS: I cos 2x sin 2x C. 2 4
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............................................................................................................................................................. i) x I x e dx ĐS: (1 ) x I x e C.
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. x e (sin x cos x) j) x I e sin x dx ĐS: I C. 2
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............................................................................................................................................................. k) I x cosx dx ĐS: I x sinx cos x C.
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............................................................................................................................................................. x x x l) I x sin dx ĐS: I 2x cos 4 sin . C 2 2 2
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............................................................................................................................................................. m) x I x e dx ĐS: x x I xe e C.
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............................................................................................................................................................. 2 2 x ln(1 x) (1 x) n) I
x ln(1 x) dx ĐS: I ln(1 x) C. 2 2 4
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............................................................................................................................................................. 2 x x sin 2x cos2x o) 2 I x sin x dx ĐS: I C. 4 4 8
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. p) 2 I ln(x 1 x ) dx ĐS: 2 2 I x ln(x 1 x ) 1 x C.
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............................................................................................................................................................. 1 x 2 x 1 1 x q) I x ln dx ĐS: I x ln C. 1 x 2 1 x
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............................................................................................................................................................. ln x ln x 1 r) I dx ĐS: I C. 3 x 2 2 2x 4x
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............................................................................................................................................................. 1 1 s) I
x sin x cosx dx ĐS: I x cos 2x sin 2x C. 4 8
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............................................................................................................................................................. 1 t) 2x I e cos 3x dx ĐS: 2x I e (3 sin 3x 2 cos 3 ) x . C 13
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............................................................................................................................................................. x dx 1 1 u) I ĐS: I x tan x ln cos x C. 1 cos 2x 2 2
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. x 1 v) 2 I x (2 cos x 1) dx ĐS: I sin 2x cos2x C. 2 4
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............................................................................................................................................................. 4 4 x ln x x w) 3 I x ln x dx ĐS: I C. 4 16
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............................................................................................................................................................. x x) I dx ĐS: I x cotx ln sin x C. 2 sin x
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............................................................................................................................................................. 1 x 1 y) 2 ( 2) x I x e dx ĐS: 2 2 ( 2) x I x e e C. 2 4
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............................................................................................................................................................. z) 2 I x ln(x 1) dx ĐS: 2 2 2 I (x 1)ln(x 1) x 1 C.
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............................................................................................................................................................. BT 14. T h g h 2 x 1 1 1 a) I ln x dx ĐS: I x ln x x C. 2 x x x
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN
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............................................................................................................................................................. b) I cos x dx ĐS: I 2 x sin x 2cos x C.
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............................................................................................................................................................. c) I sin x dx ĐS: I 2 x cos x 2sin x C.
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............................................................................................................................................................. 2 2 2 d) 3 (8 2 ) x I x x e dx ĐS: 2 (4 1) x 4 x I x e e C.
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............................................................................................................................................................. 2 1 2 x 1 2 e) 3. x I x e dx ĐS: 2 x I x e e C. 2 2
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............................................................................................................................................................. 3 1 3 x 1 3 f) 5 x I x e dx ĐS: 3 x I x e e C. 3 3
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............................................................................................................................................................. g) sin x I e sin 2x dx ĐS: sin x sin 2sin . 2 x I x e e C.
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............................................................................................................................................................. h) x I x e dx ĐS: 2 x 4 x 4 x I xe xe e C.
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............................................................................................................................................................. 1 1 i) 2 I x ln(x 1) dx ĐS: 2 2 2 I (x 1)ln(x 1) x C. 2 2
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............................................................................................................................................................. 1 ln(x 1) 1 1 x j) I dx ĐS: I ln x 1 ln C. 2 x x x x 1
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............................................................................................................................................................. k) x ln( x I e e 1) dx ĐS: ( x 1)ln( x 1) x I e e e C.
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............................................................................................................................................................. 2 ln(4x 8x 3) 2 4x 8x 3 l) I dx ĐS: 2 ln 4x 8x 3 4 ln x 1 . C 3 (x 1) 2 2(x 1)
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............................................................................................................................................................. 1 m) I 1 ln(x x 1) dx ĐS: I (x x 1)ln x x 1 x x C. 2 x
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BÀI TẬP TRẮC NGHIỆM Câu 166. M g h h ( ) x f x xe là: 2 x A. x e C B. x e x 1 C C. x e x 1 C D. x e C 2 Câu 167. M g h h 2 ( ) ( 2 ). x f x x x e là: A. (2 2). x x e B. 2 x x e C. 2 ( ). x x x e D. 2 ( 2 ). x x x e Câu 168. h h ( ) . x f x
x e . M g h F(x) f (x) hỏ F(0) 1 là: A. ( 1) x x e 1 B. ( 1) x x e 2 C. ( 1) x x e 1 D. ( 1) x x e 2
Câu 169. Cho f (x)
x sinx . Ng h f (x) là: A. x cosx C B. x sinx cosx C C. sinx x cosx C D. x cosx sinx C 2 Câu 170. Ng h h ( ) x f x xe h 2 1 2 2 2 2 A. ( ) 2 x F x e B. ( ) x F x e C. 2 ( ) 2 x F x x e D. ( ) x x F x e xe 2 x
Câu 171. Cho f (x)
lntdt . Đạ h f '(x) h ới ? 1 1 1 A. B. ln x C. 2 ln x D. 2 ln x x 2 Câu 172. M g h 3 ( ) x f x x e là: A. 3 2 ( 3 6 6) x x x x e B. 3 ( 6 6) x x x e C. 3 2 ( 3 6) x x x e D. 2 3 x x e 2 Câu 173. M g h h . x f x x e là : 2 2 1 2 2 2 A. 2 x F x e B. 2 2 x F x x e C. x F x e D. x x F x xe e 2
Câu 174. H f (x) (x
1)sinx có các nguyên hàm là: A. F(x) (x 1)cosx s inx C B. F(x) (x 1)cosx s inx C C. F(x) (x 1)cosx s inx C D. F(x) (x 1)cosx s inx C
Câu 175. H f (x)
lnx có các nguyên hàm là: 1 A. F(x) x(lnx 1) C B. F(x) C x 2 ln x C. F(x) C D. F(x) x(lnx 1) C 2 1
Câu 176. H f (x) cos x
x có các nguyên hàm là: cos x
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BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN x 1 A. 2 F(x) sin x x C B. F(x) x(1 sinx) cosx C sin x 2 C. F(x) x(1 sinx) cosx C D. F(x) x(1 sinx) cosx C
Câu 177. G i h F(x) g h f (x)
x cos 3x i F(0) 1 . Vậ F(x) là: 1 1 1 1 A. F(x) x sin 3x cos 3x C B. F(x) x sin 3x cos 3x 1 3 9 3 9 1 1 1 8 C. 2 F(x) x sin 3x D. F(x) x sin 3x cos 3x 6 3 9 9 Câu 178. Tính x xe dx A. x x F x e xe C B. x x F x e xe C C. x x F x e xe C D. x x F x e xe C Câu 179. Tính x cos xdx A. F x x sinx cosx C B. F x x sinx cosx C C. F x x sin x cosx C D. F x x sin x cosx C Câu 180. Tìm
x cos 2xdx là: 1 1 1 1
A. x sin 2x cos 2x C
B. x sin 2x cos 2x C 2 4 2 2 2 x sin 2x C. C D. sin 2x C 4 Câu 181. M g h h 2 ( ) 2 x f x x x e A. ( ) (2 2). x F x x e B. 2 ( ) x F x x e C. 2 ( ) ( ). x F x x x e D. 2 ( ) ( 2 ). x F x x x e Câu 182. M g h 2 ( ) x f x x e là 3 x A. 2 ( ) ( 2 2) x F x x x e B. F(x) x e 3 C. 2 ( ) ( 2 2) x F x x x e D. ( ) 2 x F x xe
Câu 183. Nguyên hàm F x ( ) x f x xe hỏ F 0 1 là A. ( ) 1 x F x x e 1 B. 1 x F x x e 2 C. ( ) 1 x F x x e 1 D. ( ) 1 x F x x e 2 Câu 184. K i g ? 2 x .cos x A. x sin xdx C B. x sin xdx x cosx sin x C 2
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN x cos 2x 1 C. x cosxdx x sinx cosx C D. x sin 2xdx sin 2x C 2 4 Câu 185. K i g ? 3x xe x 1 A. 3 3x xe dx e C B. x x x xe dx xe e C 3 9 2 x x x 1 C. x xe dx . x e C D. dx C 2 x x x e e e Câu 186. K i g ? 1 A. ln xdx x ln x x C B. ln xdx C x 2 2 x x 3 3 x x C. x ln xdx ln x C D. 2 x ln xdx .ln x C 2 4 3 9 Câu 187. K i g ? 3 ln x A. 2 2 ln xdx x ln x 2 x ln x x C B. 2 ln xdx C 3 ln x ln x 1 ln x ln x 1 C. dx C D. dx C 2 x x x 3 2 2 x 2x 4x Câu 188. K i g ? x x 1 A. dx C B. x x x xe dx xe e C 2x 2x 2 e 2e 4 x e 3x xe 2 x x 1 C. 3 3x xe dx e C D. 2x 2 xe dx . x e C 3 9 2 Câu 189. K i g ? 3 x 1 A. 2 x ln xdx . C 3 x 3 3 x x B. 2 x ln xdx .ln x C 3 9 C. 2 2 2 ln x 1 x dx x ln x 1 x 1 x C x e sin x cos x D. x e sin xdx C 2
Câu 190. T g h h f (x) x.sin 2x 1 x 1 A. f (x)dx .cos 2x 1 .sin 2x 1 C 2 4 2 x B. f (x)dx .cos 2x 1 C 4 x 1 C. f (x)dx .cos 2x 1 .sin 2x 1 C 2 4
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN x 1 D. f (x)dx .cos 2x 1 .sin 2x 1 C 2 2
Câu 191. T g h h f (x) x.ln 1 x 2 x A. f (x)dx C 2(x 1) 2 x 1 B. 3 f (x)dx ln 1 x x ln(1 x) C 2 6 1 1 x C. 2 2 f (x)dx x 1 .ln 1 x x C 2 4 2 2 x 1 x 1 D. 2 f (x)dx ln 1 x x ln(x 1) C 2 4 2 2
Câu 192. Ng h h I cos 2x.ln(sin x cos x)dx là: 1 1 A. F(x) = 1 sin 2x ln 1 sin 2x sin 2x C 2 4 1 1 B. F(x) = 1 sin 2x ln 1 sin 2x sin 2x C 4 2 1 1 C. F(x) = 1 sin 2x ln 1 sin 2x sin 2x C 4 4 1 1 D. F(x) = 1 sin 2x ln 1 sin 2x sin 2x C 4 4
Câu 193. Ng h h I x 2 sin 3xdx là: x 2 cos 3x 1 x 2 cos 3x 1 A. F(x) = sin 3x C B. F(x) = sin 3x C 3 9 3 9 x 2 cos 3x 1 x 2 cos 3x 1 C. F(x) = sin 3x C D. F(x) = sin 3x C 3 9 3 3
BÀI TẬP TRẮC NGHIỆM – DẠNG TOÁN KHÁC ĐỌC THÊM )
Câu 194. T g ệ h ệ h i? 2 x 6x 1 2 x 10
A. Hàm s F(x) và G(x)
là nguyên hàm c a cùng m t hàm 2x 3 2x 3 s . B. Hàm s 2 F(x) 5
2 sin x G(x) 1 cos2x là nguyên hàm c a cùng m t hàm s . x 1 C. Hàm s 2 F(x) x 2x
2 là nguyên hàm c a hàm s f (x) . 2 x 2x 2
D. Hàm s F(x)
sin x là nguyên hàm c a hàm s f (x) cos x .
Câu 195. ệ h ệ h SAI?
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TÀI LIỆU HỌC TẬP TOÁN 12 – CHẤT LƯỢNG
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN A. kf x dx k f x dx k R B. f x .g x dx f x dx. g x dx C. f x g x dx f x dx g x dx m 1 f x D. m f x f x dx C m 1 Câu 196. Để sin cos x F x a x b x e g h cos . x f x x e h gi a , b là : 1 A. a 1,b 0 B. a 0,b 1 C. a b 1 D. a b 2
Câu 197. h h f x h K . H F x g i g h h f x trên K A. Fx f x , x K B. Fx f x , x K C. f x F x , x K D. f x F x , x K
Câu 198. h h g h h h sai? A. f '(x)dx f (x) C B. Kf (x)dx K f (x)dx K 0 C. f (x) g(x)dx f (x)dx
g(x)dx D. F(x)dx f (x) C
Câu 199. T g ệ h ệ h i 3 f (x) A. 2
f '(x)f (x)dx C B.
f (x).g(x) dx
f (x)dx. g(x)dx 3 C. f (x) g(x) dx f (x)dx
g(x)dx D. kf (x)dx k f (x)dx (k là h ng s ) 2 10x 7x 2 Câu 200. N 2 f (x) (ax bx c) 2x 1 g h h g(x) 2x 1 1 h g ; thì a b c ó gi 2 A. 3 B. 0 C. 4 D. 2
Câu 201. X h a , b , c sao cho 2 g(x) (ax bx c) 2x 3 g h h 2 20x 30x 7 3 f (x) g h g ; 2x 3 2 A. a 4, b 2, c 2 B. a 1, b 2, c 4 C. a 2, b 1, c 4 D. a 4, b 2, c 1
Câu 202. Tìm nguyên hàm x 2 2 F(x) e (a tan x b tan x c) g h x 2 3 f (x) e tan x h ; 2 2
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TÀI LIỆU HỌC TẬP TOÁN 12
BÀI GIẢNG: NGUYÊN HÀM – TÍCH PHÂN x 1 2 1 x 1 2 2 A. 2 2 F(x) e tan x tan x B. 2 2 F(x) e tan x tan x 2 2 2 2 2 2 x 1 2 2 x 1 2 1 C. 2 2 F(x) e tan x tan x D. 2 2 F(x) e tan x tan x 2 2 2 2 2 2 Câu 203. N 2 ( ) x F x ax bx c e g h h 2 2 7 4 x f x x x e thì a; ; b c g hi ? A. 2; 3;1 B. 1; 3;2 C. 1; 1;1 D. 2;7; 4 Câu 204. Cho f x dx F x
C . Khi ó với a 0 , ta có f ax b dx g 1 1 A. F ax b C B. F ax b C 2a a C. F ax b C D. a.F ax b C sin cos . x x e ; x 0
Câu 205. Cho f x 1 . Nhậ é ú g ? ; x 0 1 x sin x e ; x 0 A. F x
là m t nguyên hàm c a f x 2 1 x 1; x 0 cos x e ; x 0 B. F x
là m t nguyên hàm c a f x 2 1 x 1; x 0 1 cos x e ; x 0 C. F x
là m t nguyên hàm c a f x 2 1 x ; x 0 cos x e ; x 0 D. F x
là m t nguyên hàm c a f x 2 1 x ; x 0 x a cos 3x sin 3x Câu 206. K M g h x 2 sin 3xdx 2017 h ổ g b c S . a b c g A. S 15 B. S 14 C. S 3 D. S 10 2 20x 30x 7 3 Câu 207. h h 2 f x ;F x ax bx c 2x 3 với x . Để 2x 3 2
h F x g h f x h gi , a , b c A. a 4;b 2;c 1 B. a 4;b 2;c 1 C. a 4;b 2;c 1 D. a 4;b 2;c 1
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