Tóm tắt lý thuyết, bài tập - Giải tích 1 | Trường đại học Bách Khoa, Đại học Đà Nẵng

Tóm tắt lý thuyết, bài tập - Giải tích 1 | Trường đại học Bách Khoa, Đại học Đà Nẵng được sưu tầm và soạn thảo dưới dạng file PDF để gửi tới các bạn sinh viên cùng tham khảo, ôn tập đầy đủ kiến thức, chuẩn bị cho các buổi học thật tốt. Mời bạn đọc đón xem!

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Þ Þ ¾ Þ Þ Þ ¾ Þ
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ß ß ß   ¿
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 ß 
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    
  ß  
  ¿ 
 ß  ÿ 
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  ¿ 
 ß ß  ¿   ÿ ÿ 
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  ¿  
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   ß 
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  ¿ 
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N Z Q R.
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¿   ¿ ¿  ß  ß  ß ÿ  ß    ß
¿      ß ¿ ß  ß ß ÿ ¿ 
ß ß   ¿ ¿   ß  ß ¿  ÿ  ß ß 
 ¿ ¿ 
¿  ß ÿ  ß
  ß  ß
  ß ß ¿ ¿  ¿ ß ß ¿
  ¿  ¿  ¿ ÿ ß ß  ¿ ¿ f (x) =  ¿  ¿
  ¿ 
  ß   ÿ    ß ÿ 
 ¿    ÿ ¿    ß T 6= 0(T > 0)   ß 
f (x + T) = f (x)      ß    ß T > 0  ¿
  ÿ ß    ÿ
  ÿ
 ß 
 ß  ß ÿ ß ß ÿ  
 ß  ß ß ß ÿ ß ß ¿  ÿ   ¿
   ß ß    ÿ   ß ÿ  ÿ  ¿ ß ß ÿ
 Þ ß  ß   ¿ y = a
x
, y = log
a
x    ÿ ÿ

  ß  ¿
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 ß   ß ¿  ß ¿  ß    ß
¿   ÿ  ÿ
    ß  ¿  ¿
y y y y y y y
= x
α
, = a
x
, = log
a
x, = sin x, = cos x, = tg x, = cotg x
y = arcsin x, y = arccos x, y = arctg x, y = arccotg .x
 ß   ß  ¿
  ÿ ß ß   ¿  ÿ  ÿ ÿ ÿ 
  ¿
 ¿    ÿ  ß
 y =
4
q
lg(tan x)  y = arcsin
2x
1 + x

y =
x
sin
πx
 y = arccos(2 sin x)
ß ¿
a.  = {π/4 /2+ kπ x π + kπ, k Z} b.  = {−1/3 x 1}
c.
 = {x 0, x 6Z} d.  = {
π
6
+ kπ x
π
6
+ kπ, k Z}
 ¿   ß  ß ÿ  ß

y = lg(1 2 cos x)  y = arcsin
lg
x
10
ß ¿   { y lg 3}   {π/2 y π/2}
 ¿   f (x) ¿

f
x +
1
x
= x
2
+
1
x
2
 f
x
1
+ x
= x
2
.
ß ¿
  f (x) = x
2
2 ß |x|≥2.   f (x) =
x
1
x
2
x 6= 1.
 ¿    ÿ ÿ  ß  ß   ß   ÿ
 
y = 2x + 3. y =
1 x
1
+ x
 y =
1
2
(e
x
+ e
x
)
ß ¿
 y =
1
2
x
3
2

y y= =
1 x
1 + x
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   ß ß ¿ ß 
  
y
0
=
1
2
(e
x
e
x
)   ß     ß    ¿  
ß
 ß
x 0, ÿ y =
1
2
(e e
x
+ e
x
)
x
= y ±
p
y
2
1 1x = ln(y +
p
y
2
). 
  
[ [0, +) 1, +)
x y
7→ =
1
2
(e
x
+ e
x
)
ln
(y +
q
y
2
1) y
¿  ÿ  ß
x 0  y = ln(x +
x
2
1), x 1.
 ß
x 0  ÿ    ÿ  y = ln 1.(x
x
2
1), x
 ¿    ¿ ¿ ÿ  ß

f (x) = a
x
+ a
x
(a > 0)

f (x) = ln(x +
1 x
2
)
 f (x) = sin x + cos x
ß ¿    ß     ß ¿
   ß     ß ¿
   ß    ¿  ¿
 ¿  ÿ  ¿ ¿   ß f (x)   ß  ß ¿ ß
ÿ (a, a)  ß ß ß ÿ  ¿ ß ¿ ß ÿ ß  ß ¿
ß  ß ¿
ß ¿ ß ß f (x) ¿   
f
(x) =
1
2
[ f f(x) + (x)]
| {z }
g(x)
+
1
2
[ f f(x) (x)]
| {z }
h(x)
  g(x)  ß  ß ¿  h(x)  ß  ß ¿
 ¿    ¿     ÿ  ß  ¿ 
 f (x) = A cos λx + B sin λx
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 ß   ß ¿  ß ¿  ß    ß
¿   ÿ  ÿ

f (x) = sin x +
1
2
sin 2x +
1
3
sin 3x
 f (x) = sin
2
x
 f (x) = sin(x
2
)
ß ¿  ¿ ÿ T > 0  ß   ÿ  ß    
f f(x + T) = (x)x R
A cos λ( (x + T) + B sin λ x + T) = A cos λx + B sin λx x R
A[cos λx cos λ(x + T)] + B[sin λx sin λ(x + T)] = 0 x R
2 sin
λT
2
[A sin(λx +
λT
2
) + B cos(λx +
λT
2
)] = 0 x R
sin
λT
2
= 0
T =
2kπ
λ
.
¿  ß   ¿  ß  
T =
2π
|
λ|
      ß sin x ¿  ß   2π  ß sin 2x ¿  ß
 
π  ß sin 3x ¿  ß  
2π
3
¿ f (x) = sin x +
1
2
sin 2x +
1
3
sin 3x
¿  ß   T = 2π

f (x) = sin
2
x =
1 cos 2x
2
¿  ß   T = π
 ¿ ÿ  ß   ¿  ß   T > 0 
sin sin .(x + T)
2
= (x
2
)x
 
x = 0T =
kπ, k Z, k > 0.
 
x =
πk  ß   ¿ ÿ k = l
2
, l Z, l > 0
 
x =
r
π
2
   ß  ¿
¿  ß    ¿ 
 ¿   f (x) = ax + b, f (0) = 2, f (3) = 5.  f (x)
ß ¿
 f (x) =
7
3
x 2.
 ¿   f (x) = ax
2
+ bx + c, f (2) = 0, f (0) = 1, f (1) = 5.  f (x)
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    ß ß ¿ ß 
ß ¿
 f (x) =
7
6
x
2
+
17
6
x + 1.
 ¿ 
 f (x) =
1
2
(a
x
+ a
x
), a > 0. ÿ  ¿
f f f f(x + y) + (x y) = 2 (x) (y).
 ¿  ¿ ÿ f (x) + f (y) = f (z)  ß z ¿
 f (x) = ax, a 6= 0.  f (x) = arctan x

f (x) =
1
x
 f (x) = lg
1 +
x
1 x
ß ¿
 
z = x + y   z =
x + y
1 xy
 
z =
xy
x
+ y
  z =
x + y
1 + xy
§  Þ
ß   ß   ß ß   ß ß ¿ ß ¿    
  ¿ ß ¿ ß ¿  ¿ ¿  ¿  ß  ¿ 
 ¿ ¿ ß   ß    ß ß  ß ¿  ß
 ß  ß ¿  ß   ß  ÿ   ß ß  ß ß ÿ 
   ¿ ß ¿ ¿    ¿ ß  ß ÿ ß ß ¿
   
 ß ß ß ¿
   ¿ ß ÿ
  ß ß ¿  ÿ  ¿ ß e
  ¿ ¿
 ß     ¿    ÿ  a
n

a
n
= 1 +
1
2
+
1
3
+ · · · +
1
n
 .

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  ß 
  ¿
 ¿   ß ¿ ÿ   ß 

x
n
= n
p
n
2
n  x
n
=
q
n(n + a) n  x
n
= n +
3
p
1 n
3

x
n
=
n
2
sin
nπ
2
 x
n
=
sin
2
n cos
3
n
n
ß ¿
 
1
2
 
a
2
  0       0
 ¿ 
  ß x x
n
=
n1
+
1
x
n1
, x
0
= 1.
 ÿ  ¿  {x
n
}   ß ¿ ÿ ¿
 ÿ  ¿ lim
n
x
n
= +.
 ¿ 
 u
n
= (1 +
1
n
)
n
.ÿ  ¿ {u
n
}  ß  ß   ß ¿
ß ¿  ÿ ¿ ¿ ÿ   
1
+ (1 +
1
n
) + . . . + (1 +
1
n
) (n + 1)
n+1
r
(1 +
1
n
)
n
.
(1 +
1
n
+ 1
)
n+1
(1 +
1
n
)
n
 ÿ  
u
n
= (1 +
1
n
)
n
=
n
k=0
C
k
n
.
1
n
k
k
! = 1.2 . . . k k 2
k1
2
C
k
n
.
1
n
k
=
n.(n 1 1) . . . (n k + )
k
!
.
1
n
k
<
1
k
!
1
2
k1
u
n
< 1 + 1 +
1
2
+
1
2
2
+ . . . +
1
2
k1
<
3.
 ¿ 
 s
n
= 1 +
1
1!
+ . . . +
1
n
!
ÿ  ¿ {s
n
}   ß ¿
ß ¿  lim
n
+
u
n
= lim
n
+
s
n
= e.
 ¿   lim
n+
1
+ a + . . . + a
n
1
+ b + . . . + b
n
; |a| < 1, |b| < 1.
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    ß ß ¿ ß 
ß ¿
lim
n+
1
+ a + . . . + a
n
1
+ b + . . . + b
n
= lim
n+
1 a
n+1
1
a
.
1 b
1
b
n+1
=
1 b
1 a
 ¿   lim
n
+
q
2 +
p
2 + . . . +
2  ¿ 
ß ¿
¿ u
n
=
q
2 2+
p
+ . . . +
2  u
2
n
+1
= 2 + u
n
ß ¿ ÿ  {u
n
}
 ß  ß   ß ¿ 0 u
n
2   ¿  ß ß ¿ {u
n
}  ß
 ß ß ÿ ¿ ÿ lim
n
u
n
= a, 0 < a < 2  ÿ   u
2
n
+1
= 2 + u
n
 n
 
a
2
= a + 2
¿ a = 2 lim
n
+
q
2 +
p
2 + . . . +
2 = 2
 ¿   lim
n
+
(n
n
2
1) sin .n
ß ¿ lim
n
+
(n
n
2
1) sin n = lim
n+
sin n
n
+
n
2
1
= 0   ¿ ¿
 ¿   lim
n
+
[cos( (ln n) cos ln(n + 1))].
ß ¿  
cos cos ln
(ln n) ( (n + 1)) = 2 sin
ln n + ln(n + 1)
2
. sin
ln n ln(n + 1)
2
=
2 sin
ln n(n + 1)
2
sin
ln
n
n+1
2

0
|cos cos ln(ln n) ( (n + 1))| 2
sin
ln
n
n+1
2
¿  lim
n
sin
ln
n
n+1
2
= 0     ß ¿ ¿
lim
n
+
[cos( (ln n) cos ln(n + 1))] = 0
 ¿  ÿ  ¿ lim
n+
n
2
n
= 0.
ß ¿
2
n
= (1 + 1)
n
>
n(n 1)
2
0 <
n
2
n
<
2
n
1
.
   ¿   ß ¿ ÿ 

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 ß ¿  ß 
 ¿  ÿ  ¿ lim
n+
2
n
n
!
= 0.
ß ¿  
0
<
2
n
n
!
=
2
1
.
2
2
.
2
3
. . .
2
n
< 2.
2
n
n 2
 ¿  
 lim
n
+
(
1
2
+
1
2
2
+ . . . +
n
2
n
)
 lim
n
+
(
1
3
+
1
3
2
+ . . . +
n
3
n
)
ß ¿ ÿ
 
S
n
1
2
S
n
lim
n
+
S
n
= 2.
 
S
n
1
3
S
n
lim
n
+
S
n
=
3
4
.
 ¿  ÿ  ¿ lim
n+
n
n = 1; lim
n+
n
a = 1, a > 0.
ß ¿ ¿ α
n
=
n
n 1n = (1 + α
n
)
n
>
n(n 1)
2
α
2
n
α
2
n
<
2
n
1
.  ÿ 
 ß ¿ ¿   lim
n
α
n
= 0 ¿ lim
n
n
n = 1
 ¿ a = 1, 
 ¿ a > 1, 1
n
a
n
n n > a lim
n+
n
a = 1
 ¿
a < 1, ¿ a
0
=
1
a
lim
n
a
0
= 1 lim
n+
n
a = 1.
 ¿ 
  ¿  ÿ  ¿  ß u
n
= 1 +
1
2
+ . . . +
1
n
 
 ¿  ÿ  ¿ ¿ lim
n
+
a
n
= a  lim
n+
a
1
+ a
2
+ . . . a
n
n
= a.
 ¿  ÿ  ¿ ¿ lim
n
+
a
n
= a, a
n
> 0n  lim
n+
n
a
1
.a
2
. . . a
n
= a.

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
Þ ¾ Þ
§ ¾ Þ
   ÿ  ß
    ß         ÿ ÿ  
¿    ÿ  ß ¿   ß ß  ß f (x)   ß ¿  
ß  ß F(x)  ¿  ¿ f (x) ¿ ß ¿   ¿ ¿   ß F(x) 
¿
ß  
 ß
F(x)
ÿ ß  ß
 
ÿ  ß
f (x)
 ß ¿
D
¿
F
0
(x x) = f ( )
x D

dF dx(x) = f (x)
ß     ¿   ÿ ß  ß  ß  ¿   ¿
¿ ¿ ß      ß  ¿ ÿ ¿ ¿     ÿ 
ß 
ß  
¿
F(x)
 ß   ÿ  ß
f (x)
 ¿
D

 ß
F(x) + C
  ß   ÿ  ß
f (x)
ß
C
 ß ¿ ß ¿

ÿ ¿ ß   ÿ  ß
f (x)
ß ¿ ÿ ß ¿
F(x) + C


C
 ß ¿ ß
 ¿ ß ÿ F(x) + C ß ß ¿ ¿    ÿ  ß f (x) ß ¿
ß C  ÿ   ß  

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       ß
ß  
  ¿ ß ÿ ß  ß
f (x)
 ß   
F(x) + C
ß
x D
 
C
 ß   ÿ  ß
f (x)

C
 ß ¿ ß ¿ 
  ¿ ß ÿ
f (x)dx
ÿ  ß 
Z
f (x)dx
ß ÿ
f (x)dx
ÿ ß  ß
ÿ ß ¿     ß
f (x)
ÿ ß   ß ß ¿  
¿
Z
f (x x)dx = F( ) + C ß F(x)    ÿ f (x)
  ¿ ÿ   ¿ ß
Z
f (x)dx
0
= f (x)  d
Z
f f(x)dx = (x)dx
Z
F
0
(x x)dx = F( ) + C 
Z
dF C(x) = F(x) +
Z
a f (x)dx = a
Z
f (x)dx a  ¿ ß  
Z
[
f f(x x) ± g( )
]
dx =
Z
(x x)dx ±
Z
g( )dx
  ¿ ß    ¿ ¿  ÿ   ¿ ß   ß ¿

Z
[α f f(x x) + βg( )
]
dx = α
Z
(x x)dx + β
Z
g( )dx
    ¿ ß  ß ß ¿ α, β
  ÿ   ¿  ¿
Z
x
α
dx =
x
α+1
α
+ 1
+ C, (α 6= 1)
Z
dx
x
= ln |x| + C
Z
sin xdx = cos x + C
Z
cos xdx C= sin x +
Z
dx
sin
2
x
= cotgx + C
Z
dx
cos
2
x
= tgx + C
Z
a
x
dx =
a
x
ln
a
+ C, (a > 0, a 6= 1)
Z
e
x
dx C= e
x
+
Z
dx
a
2
x
2
=
1
2
a
ln
a + x
a
x
+ C
Z
dx
x
2
+ a
2
=
1
a
arctg
x
a
+ C
Z
dx
x
2
+ α
= ln
x +
p
x
2
+ α
+ C
Z
dx
a
2
x
2
= arcsin
x
a
+ C
Z
p
a
2
x
2
dx =
1
2
x
p
a
2
x
2
+
a
2
2
arcsin
x
a
+ C
Z
p
x
2
+ adx =
1
2
h
x
p
x
2
+ a + a ln
x +
p
x
2
+ a
i
+ C

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   ¿ ß
       ¿ ß
    ß
ß  ß   ¿   ¿ ÿ ÿ     ÿ ß  ß
      ¿   ÿ    ¿ ß  ß 
  ¿     ß    ÿ   ¿ ¿
 ÿ   ¿ ß
Z
[α f (x x) + βg( )
]
dx = α
Z
f (x)dx + β
Z
g(x)dx
    ß ß ¿    ß ß ÿ   ß  ¿
  ¿ ÿ   ÿ   ¿ ß ÿ     ¿ 

 ÿ 
Z
(2x
x 3x
2
)dx = 2
Z
x
3
2
dx 3
Z
x
2
dx =
4
5
x
5
2
x
3
+ C
Z
2 sin x + x
3
1
x
dx = 2
Z
sin xdx +
Z
x
3
dx
Z
dx
x
= 2 cos x +
x
4
4
ln |x| + C
Z
dx
x
2
(1 + x
2
)
=
Z
1
x
2
1
1
+ x
2
dx =
1
x
+ arctgx + C
   ¿ ß ß ÿ  
¿  ¿
Z
f (x x)dx = F( ) + C 
Z
f (u u)du = F( ) + C   u = u(x) 
ß  ß ¿   ÿ   ß ß  ¿ ¿  ¿   ¿  x
ÿ ÿ  ¿   ¿ ß ß ÿ ß ¿   g(x)dx ß ¿
g
(x x x)dx = f (u( ))u
0
( )dx,
  f (x)  ß  ß   ß   ÿ   F(x)   
 ¿  ß 
Z
g(x)dx =
Z
f (u(x x))u
0
( )dx =
Z
f (u u(x))du = F( (x)) + C
 ß ÿ  ¿ u(x) = ax + b  du = adx   ¿
Z
f (x x)dx = F( ) + C
  
Z
f (ax + b)dx =
1
a
F(ax C+ b) +
 ÿ 

Z
sin axdx =
1
a
cos ax C+

Z
e
ax
dx =
e
ax
a
+ C

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       ß

Z Z
e
sin x
cos xdx = e
sin sinx
d(sin x) = e
x
+ C

Z
dx
cos
4
x
=
Z
(1 + tg
2
x x)d(tg ) =
tg
3
x
3
+ tgx + C

Z
x
1 + 3x
2
dx =
1
6
Z
1 + 3x
2
d(1 + 3x
2
) =
1
9
1 + 3x
2
3
+ C

I =
Z
arccos x arcsin x
1 x
2
dx
=
Z
π
2
arcsin x
arcsin arcsinxd( x)

I =
π
4
arcsin
2
x
1
3
arcsin
3
x + C
   ß ¿
   I =
Z
f (x)dx   f (x) ß  ß  ÿ ß   
    ß      ÿ ß  ß  ¿ ß
 ß ¿ x = ϕ(t)   ß ÿ ß ¿   ß ß ¿ t  ß 
ÿ   ß   ¿ 
 ß ¿ ÿ ¿
¿ x = ϕ(t)   ϕ(t)  ß  ß  ß  ¿   ÿ  
 
I =
Z
f (x)dx =
Z
f
[
φ(t t)
]
φ
0
( )dt
¿ ÿ  ß
g(t) = f
[
ϕ(t t)
]
ϕ
0
( )      G(t)  t = h(x)  
ß ÿ ÿ  ß x = ϕ(t)  
Z
g h(t t)dt = G( ) + C I = G
[
(x)
]
+ C
 ß ¿ ÿ 
¿ t = ψ(x)   ψ(x)  ß  ß  ¿   ÿ   ¿ ÿ 
f
(x x x) = g
[
ψ( )
]
ψ
0
( )    
I =
Z
f (x x x)dx =
Z
g
[
ψ( )
]
ψ
0
( )dx
¿ ÿ  ß g(t)      ß G(t) 
I = G
[
ψ(x)
]
+ C
      ¿ ß ¿   ß ¿ ß    ÿ
   ¿ ß ß ¿ ß ¿   ß ÿ ¿ ß 

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   ¿ ß
 ÿ 
   
I
1
=
Z
r
x
2
x
dx
¿
x
= 2 sin
2
t, t
0,
π
2
  ÿ
dx
= 4 sin t cos tdt,
r
x
2
x
=
s
2 sin
2
t
2
(1 sin
2
t)
= tgt
 
I
1
=
Z
r
x
2
x
dx = 4
Z
sin
2
tdt = 2t sin 2t + C
ß ¿ ¿
x
ß
t
= arcsin
p
x
2
  ÿ
I
1
=
Z
r
x
2
x
dx = 2 arcsin
r
x
2
p
2x x
2
+ C
   
I
2
=
Z
e
2x
e
x
+ 1
dx
¿
e e
x
= t
x
dx = dt
 
I
2
=
Z
t
t
+ 1
dt =
Z
1
1
t
+ 1
dt = t t ln
|
+ 1
|
+ C
ß ¿ ¿
x
 ÿ
I C
2
= e
x
ln(e
x
+ 1) +
   
I
3
=
Z
dx
1 + 4
x
¿
t
= 2
x
dt = 2
x
ln 2dx
  ß 
I
3
=
Z
dt
t
ln 2
1 + t
2
=
1
ln 2
Z
dt
t
2
+ 1
=
1
ln 2
ln(t +
p
t
2
+ 1) + C
ß ¿ ¿
x
 
I
3
=
1
ln 2
ln(2
x
+
4
x
+ 1) + C
     ÿ ¿
¿ ÿ u = u(x)  v = v(x)   ß  ¿   ÿ   ¿ ¿ 

d d(uv) = udv + vdu uv =
Z
(uv) =
Z
udv +
Z
vdu
 
Z
udv = uv
Z
vdu
   I =
Z
f (x)dx  ¿ ß ß
f (x x x x x)dx =
[
g( )h( )
]
dx = g( )
[
h( )dx udv
] =

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       ß
  ÿ  ÿ   ÿ ¿ ß   ß u = g h(x), v =
Z
(x)dx 
ß ÿ ÿ     ß ÿ ß ¿   ÿ ß 
  ß   ln x, a
x
 ß ÿ   ß ÿ  ÿ ÿ ß
   
Z
x
n
e
kx
dx kxdx;
Z
x
n
sin ;
Z
x
n
cos kxdx n   
ß ß u = x
n
   
Z
x
α
ln
n
xdx α 6= 1  n    ß ß
u = ln
n
x
  
Z
x
n
arctgkxdx;
Z
x
n
arcsin kxdx n    ß
ß u = arctgkx ¿ u = arcsin kx dx dv = x
n
 ÿ 
    ¿ ß

I
1
=
Z
ln xdx = x ln x
Z
dx C= x ln x x +

I xdx
2
=
Z
x
2
sin
¿
u = x
2
, dv = sin xdx x v = cos
 ÿ
I
2
= x
2
cos x + 2
Z
x cos xdx
¿
u = x, dv = cos xdx x v = sin
 ÿ
I
2
= x
2
cos x + 2
x sin x
Z
sin xdx
= x
2
cos x + 2x sin x + 2 cos x + C

I
3
=
Z
xe
x
dx
(
x + 1)
2
¿
u
= xe
x
; dv =
dx
(
x+1)
2
v =
1
x
+1
; du = (x + 1)e
x
dx
 ÿ
I
3
=
xe
x
x
+ 1
+
Z
e
x
dx =
xe
x
x
+ 1
+ e
x
+ C =
e
x
x
+ 1
+ C

I
4
=
Z
xe
x
dx
1 + e
x
¿
1 + e
x
= t
e
x
dx
1+e
x
= 2dt
 
I
4
= 2
Z
[
ln ln(t 1) + (t + 1)
]
dt = 2(t
1 1 2 1 1 4) ln(t ) + (t + ) ln(t + ) t + C
ß ¿ ¿
x
 
Z
xe dx
x
1 + e
x
= 2(x 2)
1 + e
x
+ +4 ln
1
1 + e
x
2x + C

I
5
=
Z
x arcsin x
1 x
2
dx
¿
u
= arcsin x; dv =
xdx
1x
2
du =
dx
1x
2
; v =
1 x
2
 ÿ
I
5
=
p
1 x
2
arcsin x +
Z
dx =
p
1 x
2
arcsin x x+ + C

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
I xdx
6
=
Z
e
x
cos 2
¿
u = cos 2x; dv = e
x
dx v = e
x
; du = 2 sin 2xdx
 ÿ
I xdx
6
= e
x
cos 2x + 2
Z
e
x
sin 2
¿
u = sin 2x; dv = e
x
dx v = e
x
; du = 2 cos 2xdx
 ÿ
I xdx I C
6
= e
x
cos 2x + 2
e
x
sin 2x 2
Z
e
x
cos 2
= e
x
cos 2x + 2e
x
sin 2x 4
6
+ 5
¿
I
6
=
e
x
5
(cos 2x + 2 sin 2x
) +
C
  ÿ     ¿    ¿ ß ÿ ß ß ¿  
¿   ÿ ÿ ÿ  ÿ   ÿ  ÿ    ß
ß   ¿  ß ß     
     ÿ ÿ ÿ
ß   ß   ÿ ÿ ÿ
 ß  ß  ¿
f
(x) =
P(x)
Q(x)
 
P(x x), Q( )
   ÿ ÿ
x
ß  ÿ ÿ ÿ  ¿ ÿ  ÿ ß ÿ ß ß 
¿ ÿ  ÿ ß ¿ ß  ß
 ÿ ÿ ÿ ÿ ÿ
¿    ÿ  P(x)  Q(x)    ÿ ß   ÿ ÿ ÿ
ß ¿
f
(x x) = H( ) +
r(x)
Q(x)
  H(x)   ÿ  r(x)  ¿      
r(x)
Q
(x)
 ß 
ÿ ÿ ÿ ÿ ÿ   ÿ  ÿ ÿ  ß  ÿ    ¿
 ¿  ß    ÿ  ÿ ÿ ÿ  ¿
r(x)
Q
(x)
  ß ÿ ¿
ß ¿ ß ÿ  ÿ   ÿ ¿ ¿ ¿  ÿ ¿   ÿ ß
ÿ ¿ ß ÿ ¿    ÿ ÿ   ß ß ¿ ß ß  ß 
ß ÿ 
  ß ß ¿ ß
¿ ÿ   ß   ß  ÿ ÿ ÿ ÿ ÿ
P(x)
Q
(x)
 ß ß
ÿ   ÿ ÿ ÿ ÿ ÿ  ¿ ß   ÿ ¿ ¿ ¿ ¿  ß ¿
    ÿ ß ¿ ß Q(x)   ÿ   ÿ ¿ ¿ ¿ ¿  
ß
Q(x) = (x x x α
1
)
a
1
...( α
m
)
a
m
(x p
2
+ p
1
x + q
1
)
b
1
...(x
2
+
n
+ q
n
)
b
n
  α
i
, p
j
, q
j
  ¿ ß a
i
, b
j
  ß   1 i m; 1 j n

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¿    ÿ Q(x) ¿ ß  ÿ (x α)
a
a  ß   
   ÿ  ÿ
P(x)
Q
(x)
¿ ß  ¿ ÿ ¿
A
i
(
xα)
i
  A
i

¿ ß  1 i a
¿    ÿ Q(x) ¿ ß ß ÿ (x
2
+ px + q)
b
b  ß 
     ÿ  ÿ
P(x)
Q(x)
¿ ß  ¿ ÿ ¿
B
j
x+C
j
(
x
2
+px+q)
j
  B
j
, C
j
  ¿ ß  1 j b
  ¿ ÿ   ÿ
P(x)
Q
(x)
   ¿ ß A
i
, B
j
, C
j
¿   ß ¿
ß ß  ¿ ß ß ¿ ß ß ÿ x
n
, n R ß  ¿  ¿ ß    ß
ß ¿ ß ¿   ß ß  ß ¿   ÿ ÿ  ¿ 

Z
Adx
x
a

Z
Adx
(x a)
k

Z
(Mx + N)dx
x
2
+ px + q

Z
(
Mx + N)dx
(
x
2
+ px + q)
m
 

Z
Adx
x
a
= A ln |x a| + C

Z
Adx
(
xa)
k
=
A
( )
k1)(xa
k1
+ C

Z
(Mx + N)dx
x
2
+ px + q
=
Z
Mt + (N Mp/2)
t
2
+ a
2
dt (a =
q
q p
2
/4, ß ¿ t = x + p/2)
=
Z
Mtdt
t
2
+ a
2
+
Z
(N Mp dt/2)
t
2
+ a
2
=
ln(t
2
+ a
2
) + (N Mp/2) arctg
t
a
+ C
=
ln(x
2
+ px + q) +
2N M p
p
4q p
2
arctg
2x + p
p
4q p
2
+ C

Z
(Mx + N)dx
(
x
2
+ px + q)
m
=
Z
Mt + ( )N Mp/2
(
t
2
+ a
2
)
m
dt
(a =
q
q p
2
/4, ß ¿ t = x + p/2)
=
Z
Mtdt
(
t
2
+ a
2
)
m
+
Z
(N Mp/2)dt
( )
t
2
+ a
2 m
  ÿ ¿
Z
Mtdt
( )
t
2
+a
2 m
=
M
2 1
(m )(t
2
+a
2
)
m1
+ C
  ÿ   ß       ÿ ¿  ß  ÿ
 ¿ ß

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   ¿ ß
 ÿ 
    ¿ ß

I
1
=
Z
x
4
x
3
+ 2x
2
2x + 1
(
x
2
+ 2)(x 1)
dx
 
x
4
x x
3
+ 2
2
2x + 1
(
x
2
+ 2)(x 1)
= x +
1
(
x
2
+ 2)(x 1)
= x +
A
x
1
+
Bx C+
x
2
+ 2
 ß ¿ ß ß  ¿
3 = (A + B)x
2
+ (C B + 2)x C
ß ¿ ß ß ÿ
x
2
, x
 ß ß ÿ   ÿ
A + B = 0
C B + 2 = 0
C = 1
A = 1
B = 1
C = 1
 
x x
4
x
3
+ 2
2
2x + 3
(
x
2
+ 2)(x 1)
= x +
1
x
1
1
2
2x
x
2
+ 2
1
x
2
+ 2
¿   ¿
I
=
x
2
2
+ ln |x 1|
ln(x
2
+ 2)
2
1
2
arctg
x
2
+ C

I
2
=
Z
2x
4
+ 10x
3
+ 17x
2
+ 16x + 5
( (
x + 1)
2
x
2
+ +2x 3)
dx
 ¿
2
x
4
+ 10 17 16 5x
3
+ x
2
+ x +
( (
x + 1)
2
x
2
+ 2x + 3)
= 2 +
2
x
+ 1
1
(
x + 1)
2
4
x
2
+ 2x + 3
 
I
= 2x + 2 ln
|
x + 1
|
+
1
x
+ 1
2
2arctg
x + 1
2
+ C
    ÿ 
   
  
Z
R(sin x, cos x)dx    ß ¿    ß ß ÿ
ÿ ÿ ß ß sin x, cos x   ß ÿ ÿ  ß ¿ ß  t = tg
t
2
 
sin
x =
2t
1
+ t
2
; cos x =
1 t
2
1
+ t
2
; tg x =
2t
1
t
2
; dx =
2dt
1
+ t
2
    ÿ  ß   ÿ  ß ÿ ¿ t

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       ß
 ÿ 
  
Z
sin x cos 2x +
1
+ sin x + cos x
dx
 ¿
Z
sin x cos 2x +
1
+ sin x + cos x
dx =
Z
d(1 + sin x + cos x)
1
+ sin x + cos x
+ 2
Z
dx
1 + sin x + cos x
¿
t
= tg
x
2
 
Z
dx
1
+ sin x + cos x
=
Z
dt
1
+ t
= ln |1 + +t
|
C
 ¿ ¿   ÿ
Z
sin x cos 2x +
1
+ sin x + cos x
dx = ln |1 + sin x + cos x
|
+ ln
1 + tg
x
2
+ C
   ¿
Z
sin cos
m
x
n
xdx   m, n   ß 
¿ m  ß   ¿  ¿ t = cos x
¿ n  ß   ¿  ¿ t = sin x
¿ m, n   ß   ¿  ÿ ÿ  ÿ ¿ ¿
sin
2
x =
1 cos 2x
2
; cos
2
x =
1 + cos 2x
2
ß  ß   ¿
Z
sin cos
k
2x
l
2xdx
 ÿ 
    ¿ ß
I xdx
1
=
Z
sin
3
x cos
2
¿
cos x = t sin xdx = dt
 
Z
sin cos
3
x
2
xdx =
Z
(1 t
2
) (t
2
dt) =
t
5
5
t
3
3
+ C =
cos
5
x
5
cos
3
x
3
+ C
I xdx
2
=
Z
sin
4
x cos
2
ÿ ÿ  ÿ ¿ ¿  
I
2
=
Z
(1 cos 2x)
2
4
1 + cos 2x
2
dx =
1
8
Z
1 cos 2x cos cos
2
2x +
3
2x
dx
I
2
=
1
8
x
sin 2x
2
Z
1+cos 4x
2
dx +
1
2
Z
(1 sin
2
2x x)d(sin 2 )
¿
I
2
=
1
8
x
2
sin 2x
2
sin 4x
8
+
sin 2x
2
sin
3
2x
6
!
+ C

CuuDuongThanCong.com https://fb.com/tailieudientucntt
   ¿ ß
ß ß  
I
2
  ÿ ÿ  ÿ ¿ ¿ ¿ ÿ ¿   
ß ¿ ÿ ¿ ¿ ÿ ß ÿ ÿ  ß ¿   ß  ÿ
sin
3
x =
3 sin x sin 3x
4
; cos
3
x =
3 cos x + cos 3x
4
 ÿ   
I
2
 
I
2
=
1
8
Z
1 cos 2x
1 + cos 4x
2
+
3 cos 2x + cos 6x
4
dx
=
1
8
x
2
sin 2x
8
sin 4x
8
+
sin 6x
24
+ C
  
Z
R(sin x, cos x)dx  ¿ ¿ ß
¿ t = cos x ¿ R( sin sinx, cos x) = R( x, cos x)
¿ t = sin x ¿ R(sin x, cos x) = R(sin x, cos x)
¿ t = tg x ¿ R( sin x, cos x) = R(sin x, cos x)
 ÿ 
  
Z
dx
sin
x cos
4
x
¿
t = cos x dt = sin xdx
 
Z
dx
sin
x cos
4
x
=
Z
dt
(
1 t
2
)t
4
=
Z
1
t
4
+
1
t
2
+
1
2 1
(t )
1
2 1
(t + )
dt
=
1
3
t
3
1
t
+
1
2
ln
t 1
t
+ 1
+ C

Z
dx
sin
x cos
4
x
=
1
3 cos
x
3
1
cos
x
+
1
2
ln
1 cos x
1
+ cos x
+ C
    ß ÿ  ÿ
    ¿
Z
R(x,
α
2
± x x v
2
)dx
Z
R( ,
x
2
α
2
)dx   R(u, )  
 ß ÿ ÿ
¿ x = α tg t ß ß  
Z
R(x x,
α
2
+
2
)dx
¿ x = α sin t ¿ x = a cos t ß ß  
Z
R(x,
α
2
x
2
)dx
¿ x =
α
cos
t
¿ x =
α
sin
t
ß ß  
Z
R(x,
x
2
α
2
)dx

CuuDuongThanCong.com https://fb.com/tailieudientucntt
       ß
  ß    ÿ  ß ÿ  ÿ  ß ÿ  ß ß 
ß ¿    ¿ 
Z
dx
x
2
+α
= ln
x +
x
2
+ α
+ C
Z
dx
a
2
x
2
= arcsin
x
a
+ C
Z
a
2
x
2
dx =
1
2
x
a
2
x
2
+
a
2
2
arcsin
x
a
+ C
Z
x
2
+ adx =
1
2
h
x
x
2
+ +a a ln
x +
x
2
+ a
i
+ C
 ÿ 
    

Z
(1 x
2
)
3
2
dx
¿
x
= sin t, t
π
2
,
π
2
dx = cos tdt,
1 x
2
= cos t

Z
(1 x
2
)
3
2
dx =
Z
dt
cos
2
t
= tg t + C = tg(arcsin x) + C

Z
dx
x
2
1+x
2
¿
x = tgt dx =
dt
cos
2
t
 
Z
dx
x
2
1 + x
2
=
Z
cos tdt
sin
2
t
=
1
sin
t
+ C =
1
sin arctg
( x)
+ C
     ¿
Z
R
x,
ax b+
cx
+d
m/n
, ...,
ax b+
cx
+d
r/s
dx
¿
ax b+
cx
+ d
= tk ß k  ß  ß ¿ ÿ  ß ß   ß ¿ ÿ ß ß t
R   ÿ ß

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§ Þ
 ß     ß
ß  
¿ ÿ  ß
f (x)
 ß  ß ¿ 
[a, b]

[a, b]

n
¿ ß
[x x
i
,
i+1
]
ß  ¿
a = x x
0
<
1
< . . . < x
n
= b
 ß ¿
[x
i
, x
i+1
]
 ß ß
ξ
i
[x x
i
,
i+1
]
  ¿ ß ÿ
S
n
=
n1
i=0
f (ξ
i
) 4 x
i
ß
4 x
i
= x
i+1
x
i

ß ÿ
S
n
ÿ ß  ß   ß
λ = max
1
in
4x
i
¿ ß ¿ ß ¿ ÿ
¿
I = lim
λ
0
S
n
 ÿ ß    ¿
[a, b]
  ÿ ß  
ß ß
ξ
i

I
ÿ ß     ß ÿ  ß
f (x)

[a, b]
  ß 
Z
b
a
f (x)dx
 ß ÿ     ß
f (x)
¿  
[a, b]
 
 ß       ß
f (x)
 ¿ 
[a, b]
ÿ 
 ¿ ¿
a < b
 ß ¿
b < a
 ß 
Z
b
a
f (x)dx :=
Z
a
b
f (x)dx
 
a = b
 ß 
Z
b
a
f (x)dx = 0
   ¿ ¿ 
ß  
ß ß ¿  ÿ ß  ß ß ¿
f (x)
¿  
[a, b]

lim
λ
0
(S
s) = 0
 
S =
n+1
i=1
M
i
4 x
i
, s =
n+1
i=1
m
i
4 x
i
M
i
= sup
x[x x
i
,
i+1
]
f (x), m
i
= inf
x
[x
i
,x
i+1
]
f (x)
 ÿ ß      ß ÿ  ÿ  ß  
ß  
¿
f (x)
 ÿ 
[a, b]

f (x)
¿  
[a, b]
ß  
¿
f (x)
ß ¿ 
[a, b]
  ß ß ß  ¿ 
[a, b]

f (x)
¿  
[a, b]
ß  
¿
f (x)
ß ¿   ß 
[a, b]

f (x)
¿  
[a, b]

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       ß
   ¿ ÿ    ß
  ¿ ¿    ¿        ¿
Z
b
a
f (x)dx 
ß  f (x) ÿ ¿ ¿ ¿   [a, b]
 ¿ 
b
Z
a
[α f ( (x) + βg x)]dx = α
b
Z
a
f (x)dx + β
b
Z
a
g(x)dx
 ¿ 
 ¿  [a a, b], [ , c c], [b, ] ¿ f (x) ¿   ¿  ß  ß ¿
  ¿   ¿  ¿ 
b
Z
a
f (x)dx =
c
Z
a
f (x)dx +
b
Z
c
f (x)dx
 ¿  ¿ ¿ a < b  
 ¿ f (x) 0, x [a, b] 
Z
b
a
f (x)dx 0
 ¿ f (x x) g( )x [a, b] 
Z
b
a
f (x)dx
Z
b
a
g(x)dx
 ¿ f (x) ¿   [a, b]  | f (x)| ¿   [a, b] 
|
b
Z
a
f (x)dx |≤
b
Z
a
| f (x) | dx
 ¿ m f (x) M, f orallx [a, b] 
m(b a)
b
Z
a
f (x)dx M(b a)
 ¿ ß   ÿ ¿
¿ ÿ f (x) ¿   [a, b]  m f (x) M, x [a, b]   ß ¿ µ  
b
Z
a
f (x)dx = µ(b a), m µ M.
¿ ß ¿ f (x)  ÿ  [a, b]  ß ¿ c [a, b]  
b
Z
a
f f(x)dx = (c)(b a).

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    ß
 ¿ ß   ÿ 
¿ ¿
 f (x)  f (x x)g( ) ¿   [a, b]
 m f (x) M, x [a, b]
 g(x)  ß ¿  [a, b]
 
b
Z
a
f (x x)g( )dx = µ
b
Z
a
g(x)dx, m µ M.
¿ ß ¿ f (x)  ÿ  [a, b]  ß ¿ c [a, b]  
b
Z
a
f f(x x)g( )dx = (c)
b
Z
a
g(x)dx.
   ß ¿   ß   
¿ ÿ f (x)  ß  ¿   [a, b]   ß ß x [a, b]  f  ¿ 
 [a, x]   ß  ß F(x) =
x
Z
a
f (t)dt
ß   
¿
f (t)
¿  
[a, b]

F(x)
 ÿ 
[a, b]

¿
f
 ÿ ¿
x
0
[a, b]

F(x)
 ¿  ¿
x
0

F
0
(x
0
) = f (x
0
)
ß    ÿ 
¿
f (x)
 ÿ  ¿ 
[a, b]

F(x)
 ß   ÿ
f (x)

b
Z
a
f (x)dx = F F(b) (a).
        ß
 ÿ ÿ  ÿ   ÿ ¿
¿ ÿ u(x x), v( )    ß  ¿   ÿ  [a, b]  
b
Z
a
udv = uv|
b
a
b
Z
a
vdu

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       ß
 ÿ ÿ   ß ¿ ß
ß   ß ¿ x := ϕ(t)

I =
b
Z
a
f (x)dx
ß
f (x)
 ÿ 
[a, b]
ÿ ß  ß ¿
x = ϕ(t)
¿  ß ß 
 ϕ(t)
 ¿   ÿ 
[a, b]
 ϕ(a) = α; ϕ(b) = β.


t
¿  
[α, β]
ÿ
α
¿
β

x = ϕ(t)
¿   ÿ ÿ
a
¿
b
     ÿ
b
Z
a
f (x)dx =
β
Z
α
f
[ϕ(t t)]ϕ
0
( )dt.
ß   ß ¿ t := ϕ(x)
¿ ÿ   ¿   ¿
I =
b
Z
a
f
[ϕ(x x)].ϕ
0
( )dx
 
ϕ(x)
¿   ß ¿   ¿   ÿ 
[a, b]
 
b
Z
a
f
[ϕ(x x)].ϕ
0
( )dx =
ϕ(b)
Z
ϕ(a)
f (t)dt.
 ÿ ÿ    ß  ¿
 ß ß  ¿
¿   ¿  ÿ   
     ÿ 
x
Z
a
f
(t)dt
0
x
= f (x) 
g(x)
Z
a
f
(t)dt
0
x
=
f ( ( (g x)).g
0
x) 
 ÿ     ¿  ß     ÿ  ÿ   ÿ 
ÿ ¿  ÿ  ÿ

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    ß
 ¿ 
  ¿ 
a
)
d
dx
y
Z
x
e
t
2
dt b)
d
dy
y
Z
x
e
t
2
dt c)
d
dx
x
3
Z
x
2
dt
1 + x
4
ß ¿

d
dx
y
Z
x
e
t
2
dt =
d
dx
x
Z
y
e
t
2
dt = e
x
2
(  y  ¿ ß)

d
dy
y
Z
x
e
t
2
dt = e
y
2
(  x  ¿ ß)

d
dx
x
3
Z
x
2
dt
1 + x
4
=
d
dx
x
2
Z
a
dt
1 + x
4
+
d
dx
x
3
Z
a
dt
1 + x
4
=
2x
1 + x
8
+
3x
2
1 + 12x
2
¿   ß ¿ ÿ  ß ÿ   ÿ   ¿  ÿ
  
 ¿ 
 ß ¿
a)A = lim
x0
+
sin x
Z
0
tg tdt
tg x
Z
0
sin tdt
b)B = lim
x+
x
Z
0
(
arctg t)
2
dt
x
2
+ 1
ß ¿
 ¿  lim
x0
+
sin x
Z
0
tg tdt = lim
x0
+
tg x
Z
0
sin tdt = 0   ÿ  ¿ 

lim
x0
+
sin x
Z
0
tg tdt
!
0
tg x
Z
0
sin tdt
!
0
= lim
x
0
+
p
tg(sin x). cos x
p
sin(tg x).
1
cos
2
x
= =1A 1

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       ß
 ¿  lim
x+
x
Z
0
(
arctg t)
2
dt = lim
x
+
x
2
+ 1 =   ÿ  ¿ 
 
lim
x+
x
Z
0
(
arctg t)
2
dt
!
0
x
2
+ 1
0
= lim
x+
(
arctg x)
2
x
x
2
+1
=
π
2
4
B =
π
2
4
¿  ÿ ÿ  ÿ ß   ß  ß ¿ ÿ ß ß  ß
¿ ß
¿  ÿ  ÿ  ß   
S
n
=
n1
i=0
f (ξ
i
) 4 x
i
ß 4 x x x
i
=
i+1
i
¿    ¿ [a, b]  n ¿  ß  ¿  ß  ¿ a =
x x
0
<
1
< . . . < x
n
= b   x
i
= a + (b a)
i
n

S
n
=
b a
n
n1
i=0
f (ξ
i
) ß ξ
i
[x
i
, x
i+1
]
  ¿  f (x) ¿   [a, b]  ß ξ
i
= x
i
 ÿ  ÿ
lim
n
b a
n
"
n1
i=0
f
a +
b a
n
.i
!#
=
b
Z
a
f (x)dx 
 ¿ ß ξ
i
= x
i+1
 ÿ  ÿ
lim
n
b a
n
"
n
i=1
f
a +
b a
n
.i
!#
=
b
Z
a
f (x)dx 
 ¿ 
 ß       ß   ß ¿

A = lim
n
h
1
n
α
+
1
n
α+β
+
1
n
α+2β
+ · · · +
1
n n
α+( 1)β
i

B = lim
n
1
n
q
1 +
1
n
+
q
1 +
2
n
+ · · · +
p
1 +
n
n
ß ¿

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    ß
 ¿
A = lim
n
1
n
"
1
α
+
1
α
+
β
n
+
1
α
+
2β
n
+
· · · +
1
α
+
(n1)β
n
#
 ÿ  ÿ  ß a = 0, b = 1, f (x) =
1
α
+βx
 ÿ
A =
1
Z
0
1
α
+ βx
dx =
1
β
ln
α + β
α
¿  ÿ  ÿ  ß a = 0, b = 1, f (x) =
1
α
+βx
 ÿ
A
0
= lim
n
h
1
n
α + β
+
1
n
α + 2β
+ · · · +
1
n
α + nβ
i
= A =
1
β
ln
α + β
α
  ÿ  ÿ  ß
a = 0, b = 1, f (x) =
1 + x  ÿ
B =
1
Z
0
1 + xdx =
2
3
(2
2 1)
¿  ÿ  ÿ  ß
a = 0, b = 1, f (x) =
1 + x  ÿ
B
0
= lim
n
1
n
1 +
r
1 +
1
n
+ · · · +
r
1 +
n 1
n
!
= B =
2
3
(2
2 1)
 ¿ 

lim
n
1
n
n
q
(2n)!
n
!
!
¿      ß  ÿ 
 ¿ 
   
a.
e
Z
1
| ln x | (x + 1)dx d.
2
Z
0
sin
2
x cos x
( )
1 + tg
2
x
2
dx
b.
e
Z
1
(
x ln x)
2
dx e.
3
Z
0
arcsin
r
x
1
+ x
dx
c.
1
Z
0
(
x
3
2x + 5)e
x
2
dx f .
π
2
Z
0
cos
n
x cos nxdx
ß ¿

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       ß
   a, b, c ß ¿ ¿     ÿ ¿  ß  
I
a
=
e
2
+ 5
4
, I
b
=
5e
3
2
27
, I
c
= 98
144
e

I
d
=
2
Z
0
sin
2
x cos x
( )
1 + tg
2
x
2
dx
=
2
Z
0
sin
2
x. cos . cosx
4
xdx
=
2
Z
0
sin sin
2
x.(1
2
x)d(sin x)
=
sin
3
(2)
3
2 sin
5
(2)
5
+
sin
7
(2)
7

I
e
=
3
Z
0
arcsin
r
x
1
+ x
dx
=
x arcsin
r
x
1
+ x
3
0
3
Z
0
x
.
1
q
1
x
1+x
.
1
2
q
x
x+1
.
1
(
x + 1)
2
dx
=
π
1
2
3
Z
0
x
x
+ 1
dx
=
π
1
2
3
Z
0
t
t
2
+ 1
.2tdt ¿
x = t)
= π
3
Z
0
1
1
t
2
+ 1
dt
=
π
h
(t arctg t)
3
0
i
=
4π
3
3

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    ß

I
n
=
π
2
Z
0
cos
n
x cos nxdx
=
1
n
π
2
Z
0
cos
n
xd sin nx
=
1
n
cos
n
x sin nx
π
2
0
+
1
n
π
2
Z
0
sin
nx.n. cos
n1
x. sin xdx
=
π
2
Z
0
sin
nx. cos . sin
n1
x xdx
 
2I
n
=
π
2
Z
0
cos
n
x cos nxdx +
π
2
Z
0
sin
nx. cos
n1
x. sin xdx
=
π
2
Z
0
cos cos
n1
x (n 1)xdx
= I
n1
¿    ß  
I
n
=
1
2
n
.I
0
=
π
2
n+1
 ¿ 

I
n
=
π
2
Z
0
sin
n
xdx, J
n
=
π
2
Z
0
cos
n
xdx
¿  ÿ   ¿ ÿ  
 ¿ 
ÿ  ¿ ¿
f (x)
 ÿ 
[0, 1]

a/
π
2
Z
0
f (sin x)dx =
π
2
Z
0
f (sin x)dx, b/
π
Z
0
x f
(sin x)dx =
π
2
π
Z
0
f (sin x)dx
ß ¿    ¿ ß   ¿ t =
π
2
x    ¿ t = π x
 ¿ 
 ÿ ¿ ¿ ÿ  ¿   ÿ 
π
2
Z
0
sin x
sin x +
cos x
dx =
π
2
Z
0
cos x
sin x +
cos x
dx
=
π
4

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       ß
 ¿ 
¿ ÿ
f (x)
 ÿ 
[a, a]( )a > 0
 ÿ 
I =
a
Z
a
f
(x)dx =
0
¿
f (x)
  ß ¿ 
[a, a]
2
a
Z
0
f (x)dx
¿
f (x)
  ß ¿ 
[a, a]
 ¿ 

f (x)
 ÿ ¿ 
[a, a]
ÿ 
a
Z
a
f (x)dx
1
+ b
x
=
a
Z
0
f (x)dx
ß
0 b 6= 1
 ÿ 
I
1
=
1
Z
1
1
(
x
2
+ +1)(e
x
1)
dx I,
2
=
π
2
Z
π
2
2
x
cos 2x
2002
x
+ 2
x
dx, I
3
=
π
2
Z
π
2
x
2
| sin x |
1
+ 2
x
dx
 ¿ 
ÿ 
b
Z
a
x
m
(a + b x)
n
dx =
b
Z
a
x x
n
(a + b )
m
dx
 ÿ 
I
n
=
1
Z
0
x
2
(1 x)
n
dx
 ÿ 
n
k=0
(1)
k
C
k
n
.
1
k
+ 3
=
1
n
+ 1
2
n
+ 2
+
1
n + 3
¿  ÿ   ¿ ¿ ÿ  
 ¿ 

f (x x), g( )
   ß ¿  
[a, b]
 
f
2
(x), g
2
(x)

¿  
[a, b]
ÿ  ¿ ¿ ÿ 
(a < b)
b
Z
a
f
(x x)g( )dx
!
2
b
Z
a
f
2
(x)dx
!
.
b
Z
a
g
2
(x)dx
!
¿ ¿ ÿ 
ß ¿  ß ÿ
 ¿
b
Z
a
f
2
(x)dx =
b
Z
a
g
2
(x)dx = 0 
0
b
Z
a
f
(x x)g( )dx
b
Z
a
| f (x x)g( )|dx
b
Z
a
f
2
(x x) + g
2
( )
2
dx = 0
   ¿ = ¿ 

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    ß ß ¿

z
0
x
= y cos
x
y
; z
0
y
= 2y sin
x
y
x cos
x
y
.

z
0
x
= y
3
x
y
3
1
; z
0
y
= 3y
2
ln x.x
y
3

z
0
x
=
1
x
2
y
2
x
2
+y
2
+ 1
x
s
x
2
y
2
x
2
+ y
2
!
=
y
2
x
p
x
4
y
4
z
0
y
=
1
x
2
y
2
x
2
+y
2
+ 1
y
s
x
2
y
2
x
2
+ y
2
!
=
y
p
x
4
y
4

u
0
x
= y
z
x
y
z
1
; u
0
y
= x
y
z
zy
z1
. ln x; u
0
z
= x
y
z
y
z
ln y ln x

u
0
x
= e
1
x
2
+y
2
+z
2
.
2x
(
x
2
+ y
2
+ z
2
)
2
; u
0
y
= e
1
x
2
+y
2
+z
2
2y
(
x
2
+ y
2
+ z
2
)
2
; u
0
z
= e
1
x
2
+y
2
+z
2
2z
(
x
2
+ y
2
+ z
2
)
2
.
 ¿  ¿  ÿ  ÿ ÿ ß ¿  ÿ ÿ  ¿   ÿ 
 ß f
(
x y,
)


f
(
x y,
) =
x
arctg
y
x
2
¿ x 6= 0
0 ¿ x = 0

f
(
x y,
) =
x ysin y sin x
x
2
+ y
2
¿ (x y,
)
6=
(
0, 0
)
0 ¿
(
x, y
) =
(
0, 0 .
)

CuuDuongThanCong.com https://fb.com/tailieudientucntt
 ¿     
ß ¿   ß ¿  ß  ÿ ß ß
(
x, y
)
6=
(
0, y
)

x = 0 
x arctg
y
x
2
π
2
|x|  lim
x
0
x. arctg
y
x
2
= 0 = f
(
0, y
)
. ¿ f
(
x, y
)

ÿ  R
2
ß x 6= 0  ¿   ß ¿   ÿ
z
0
x
= arctg
y
x
2
2x
2
y
2
x
4
+ y
4
, z
0
y
=
2x
3
y
x
4
+ y
4
 ¿ x = 0
f
0
x
(0, y
) =
lim
h0
f f
(
h, y
)
(
0, y
)
h
= arctg
h
y
2
=
0, 0y =
π
2
, y 6= 0
f
0
y
(
0, y
) =
lim
k0
f
(
0, y + k
)
f
(
0, y
)
k
= lim
k
0
0 = 0
¿  ¿
f
0
x
(x, y
)
 ÿ  R
2
\
(
0, 0
)
f
0
y
(x, y
)
 ÿ  R
2
  ß  ÿ  R
2
\
(
0, 0
)
 ¿ (0, 0) 
0
x sin y ysinx
x
2
+ y
2
=
xy
x
2
+ y
2
sin y
y
sin x
x
1
2
sin y
y
sin x
x

lim
x0
y
0
x sin y ysinx
x
2
+ y
2
= 0
¿ f (x y, )  ÿ  R
2
 ¿ 
¿ ÿ z = y f
x
2
y
2
, ß  f   ß ¿  ÿ  ¿ ß ß
 ß z ß ÿ   ¿ 
1
x
z
0
x
+
1
y
z
0
y
=
z
y
2
ß ¿  
z
0
x
= y f
x
2
y
2
.2x, z
0
y
= f
x
2
y
2
+ y. f
x
2
y
2
.
(
2y
)

1
x
z
0
x
+
1
y
z
0
y
=
f
x
2
y
2
y
=
z
y
2
 ¿   ¿  ÿ  ß ÿ  

z = e
u
2
2v
2
, u = cos .x, v =
p
x
2
+ y
2

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    ß ß ¿

z = ln
u
2
+ v
2
, ,u = xy v =
x
y
.
 z = arcsin .
(
x y
)
, x = 3 4t, y = t
3
ß ¿   
(
u
0
x
= sin x
u
0
y
= 0
;
v
0
x
=
x
x
2
+y
2
v
0
y
=
y
x
2
+y
2
;

z
0
x
= e
cos x
2
2
(
x
2
+y
2
)
[ sin 2x 4x] .
z
0
y
= e
cos x
2
2
(
x
2
+y
2
)
[4y] .
  
(
u
0
x
= y
u
0
y
= x
;
(
v
0
x
=
1
y
v
0
y
=
x
y
2

z
0
x
=
2
x
, z
0
y
=
2
y
4
1
y
(
y
4
+ 1
)
  
(
x
0
t
= 3
y
0
t
= 12t
2

z
0
t
=
1
q
1
(
x y
)
2
3 12t
2
 ¿      ¿ ÿ   ß

z = sin
x
2
+ y
2
.  z = ln tg
y
x

z = arctg
x + y
x
y
 u = x
y
2
z
. 
ß ¿ 
dz
= cos
x
2
+ y
2
(
2xdx + 2ydy
)

dz
=
2
sin
2y
x
.
xdy ydx
x
2
.

CuuDuongThanCong.com https://fb.com/tailieudientucntt
 ¿     

dz
=
(x y
)
dx +
(
x + y
)
dy
(
x y)
2
+ (x + y)
2
.

du
= x
y
2
z
y
2
z
x
dx + 2yz ln xdy + y
2
ln xdz
.
 ¿   ¿ 
 A =
3
q
(1, 02)
2
+
(
0, 05)
2
 B = ln
3
1, 03 +
4
0, 98 1
ß ¿    f
(
x y,
) =
3
p
x
2
+ y
2
, x = 0, 02; y x= 0, 05; = 1; y = 0.  
f
0
x
=
1
3
(
x
2
+ y
2
)
2/3
2x; f
0
y
=
1
3
(
x
2
+ y
2
)
2/3
2y
 
f
(
1 + x, 0 + y
)
f
(
1, 0
) +
f
0
x
(
1, 0
)
x + f
0
y
(
1, 0
)
y = 1 +
2
3
.0, 02 0.0, 05+ = 1, 013.
  
f
(
x, y
) =
ln
3
x +
4
y 1
; x = 1; 1;y = x = 0, 03; y = 0, 02
 
f
0
x
=
1
3
x +
4
y 1
.
1
3
x
2
3
;
f
0
y
=
1
3
x +
4
y 1
.
1
3
y
3
4
 
f
(
1 + x, 1 + y
)
f
(
1, 1
) +
f
0
x
(
1, 1
)
x + f
0
y
(
1, 1
)
y = 0 +
1
3
.0, 03 +
1
4
(0, 02) = 0, 005.
 ¿   ¿  ÿ   ß ¿  ß ß    

x
3
y y
3
x = a
4
;  y
0
 arctg
x + y
a
=
y
a
;  y
0

x y+ + z = e
z
;  z
0
x
, z
0
y
 x
3
+ y
3
+ z
3
3xyz = 0,  z
0
x
, z
0
y
ß ¿
   ß ¿
F
(
x, y x
) =
3
y x y
3
a
4
= 0  F
0
x
= 3x
2
y y
3
; F
0
y
= x
3
3y
2
x. ¿
y
0
=
F
0
x
F
0
y
=
3x
2
y y
3
x
3
3y
2
x

CuuDuongThanCong.com https://fb.com/tailieudientucntt
    ß ß ¿
   ß ¿ F
(
x, y
) =
arctg
x+y
a
y
a

F
0
x
=
1
a
1
+
(
x+y
a
)
2
=
a
a
2
+
(
x+y
)
2
F
0
y
=
a
a
2
+
(
x+y
)
2
1
a
=
a
2
a
2
(
x+y
)
2
a
(
a
2
+
(
x+y
)
2
)

y
0
=
a
(
x + y)
2
.
   ß ¿
F
(
x, y, z
) =
x + y + z e
z
 F
0
x
= 1; F
0
y
= 1; F
0
z
= 1 e
z

z
0
x
=
1
1
e
z
; z
0
y
=
1
1 e
z
   ß ¿
F
(
x y x,
) =
3
+ y
3
+ z
3
3xyz = 0  F
0
x
= 3x
2
3yz; F
0
y
= 3y
2
3xz; F
0
z
=
3z
2
3xy 
z
0
x
=
3yz 3x
2
3
z
2
3xy
; z
0
x
=
3xz 3y
2
3z
2
3xy
 ¿   u =
x+z
y
+z
 u
0
x
, u
0
y
¿ ¿ z   ß ¿ ÿ x, y  ß ß
  z.e
z
= x.e
x
+ y.e
y
ß ¿
  ß F
(
x, y, z
) =
ze xe ye
z
x
y
= 0 
F
0
x
=
(
e
x
+ xe
x
)
F
0
y
=
(
e
y
+ ye
y
)
F
0
z
= e
z
+ ze
z

u
0
x
=
(1 + z
0
x
) . (y + z
)
(
x + z
) (
z
0
x
)
(
y + z)
2
=
1
+
e
x
+xe
x
e
z
+ze
z
(
x + z
)
e
x
+xe
x
e
z
+ze
z
(
y + z)
2
u
0
y
=
(
x + z
)
.
1 + z
0
y
(
y + z
)
z
0
y
( )
y + z
2
=
(x + z) .
1 +
e
y
+ye
y
e
z
+ze
z
(
y + z
)
e
y
+ye
y
e
z
+ze
z
(
y + z)
2
 ¿   ¿  ÿ   ß ¿ y(x x), z( )  ß ß ß
x + y + z = 0
x
2
+ y
2
+ z
2
= 1
ß ¿ ¿ ¿   ¿ ÿ    ÿ ß  
1
+ y
0
x
+ z
0
x
= 0
2
x + 2yy
0
x
+ 2zz
0
x
= 0

y
0
x
=
z x
y z
z
0
x
=
x y
y z

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 ¿     
 ¿    z
2
+
2
x
=
p
y
2
z
2
 ß  ¿ z = z
(
x, y
)
ÿ 
¿
x
2
z
0
x
+
1
y
z
0
y
=
1
z
ß ¿   ß F
(
x, y, z
) =
z
2
+
2
x
p
y
2
z
2

F
0
x
=
2
x
2
F
0
y
=
y
y
2
z
2
F
0
z
= 2z +
z
y
2
z
2

z
0
x
=
2
x
2
2
z +
z
y
2
z
2
z
0
y
=
y
y
2
z
2
2
z +
z
y
2
z
2
ÿ   
x
2
z
0
x
+
z
0
y
y
=
1
z
.
 ¿    ¿   ¿  ÿ   ß 

z =
1
3
q
(x
2
+ y
2
)
3
 z = x
2
ln
x
2
+ y
2
 z = arctg
y
x
ß ¿
  
z
0
x
= x
p
x
2
+ y
2
z
0
y
= y
p
x
2
+ y
2

z
00
xx
=
p
x
2
+ y
2
+ x
2x
2
p
x
2
+ y
2
=
2x
2
+ y
2
p
x
2
+ y
2
z
00
yy
=
p
x
2
+ y
2
+ y
2y
2
p
x
2
+ y
2
=
x
2
+ 2y
2
p
x
2
+ y
2
z
00
xy
=
2xy
2
p
x
2
+ y
2
=
xy
p
x
2
+ y
2
  
z
0
x
= 2x ln
(
x + y
) +
x
2
x + y
z
0
y
=
x
2
x + y

z
00
xx
= 2 ln
(
x + y
) +
2x
x
+ y
+
2x
(
x + y x
)
2
(
x + y)
2
z
00
xy
=
2x
x
+ y
+
x
2
(
x + y)
2
z
00
yy
=
x
2
(
x + y)
2
  
z
0
x
=
1
1
+
y
x
2
.
y
x
2
=
y
x
2
+ y
2
z
0
y
=
1
1
+
y
x
2
1
x
=
x
x
2
+ y
2

z
00
xx
=
2xy
(
x
2
+ y
2
)
2
z
00
xy
=
x
2
+ y
2
+ y.2y
(
x
2
+ y
2
)
2
=
y
2
x
2
(
x
2
+ y
2
)
2
z
00
yy
=
2xy
(
x
2
+ y
2
)
2

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    ß ß ¿
 ¿     ¿  ÿ   ß 

z = xy
2
x
2
y  z =
1
2
(
x
2
+ y
2
)
ß ¿
   dz =
y
2
2xy
dx +
2xy x
2
dy 
d
2
z = 2y
(
dx
)
2
+ 4
(
y x
)
dxdy dy+
(
2y
) ( )
2
   dz =
x
2
(
x
2
+y
2
)
2
dx +
y
2
(
x
2
+y
2
)
2
dy 
d
2
z =
y
2
3x
2
(
x
2
+ y
2
)
3
(dx)
2
4xy
(
x
2
+ y
2
)
3
dxdy +
x
2
3y
2
(
x
2
+ y
2
)
3
(dy)
2
§ þ Þ þ Þ Þ ¾ Þ
 ÿ ß ÿ 
ß  
  ß
z = f (x y, )
 ß  ß ß
D

M D
0
(x
0
, y
0
)
  ¿  ß
f (x y, )
¿
ÿ ß
¿
M
0
¿ ß ß ß
M
  ¿   ÿ
M
0
 
M
0
ß ß
f f(M) (M
0
)
 ¿  ß
¿
f f(M) (M
0
) > 0
 ß  ¿   ÿ
M
0

M
0
ÿ ß 
ÿ ß
ÿ  ß
f
¿
M
0
¿
f f(M) (M
0
) < 0
 ß  ¿   ÿ
M
0

M
0
ÿ ß 
ÿ ¿
ÿ  ß
f
¿
M
0
 ¿ ¿    ÿ ÿ   ß 
p
= f
0
x
(M), q = f
y
(M), r = f
xx
(M), s = f
xy
(M), t = f
yy
(M)
ß  
¿  ß
f (x y, )
¿ ÿ ß ¿
M
 ¿   ¿  
p =
f
0
x
(M), q = f
y
(M)
ß ¿   ¿   ¿ ¿ 
ß  
¿ ÿ  ß
z = f (x y, )
  ¿   ¿ ¿   ÿ 
ß  ¿   ÿ
M
0
(x y
0
,
0
)
¿ ÿ ¿
M
0
 
p = q = 0
 
 ¿
s
2
rt < 0

f (x y, )
¿ ÿ ß ¿
M
0
  ÿ ß ¿
r > 0
 ÿ ¿ ¿
r < 0

CuuDuongThanCong.com https://fb.com/tailieudientucntt
 ÿ ß ÿ  ß ß ¿ ß
 ¿
s
2
rt > 0

f (x y, )
 ¿ ÿ ß ¿
M
0
  ¿ s
2
rt = 0   ¿ ¿ ÿ ß  ß ß M
0
  ß  ÿ ß
  ß   ß ÿ   ¿  ß  ß   M
0
 ¿ 
ÿ ß   ¿   ß f f(M) (M
0
) ¿   ß ¿  ß 
¿   ÿ M
0
   ÿ ß  ÿ ¿
 ¿   ÿ ß ÿ   ß 
 z = x
2
+ xy + y
2
+ x y + 1 z = x + y x.e
y

z = 2x x
4
+ y
4
2
2y
2
 z = x
2
+ y
2
e
(
x
2
+y
2
)
ß ¿
  ß  
p
= z
0
x
= =2x + +y 1 0
q
= z
0
y
= x y+ 2 1 = 0
x = 1
y = 1
¿  
M
(
1, 1
)
 ß ß ¿  ¿
 
A = z
00
xx
(M) = 2; B = z
00
xy
(M) = 1; C = z
00
yy
(M) = 2  B
2
AC = 1 4 = 3 <
0. ¿  ß ¿ ÿ ß ¿ M   A > 0  M  ß ÿ ß
  ß  
p = 1 e
y
= 0
q = 1 xe
y
= 0
x = 1
y = 0
¿  ß  ß ß ¿  ¿
M
(
1, 0
)
  A = z
00
xx
(M) = 0; B = z
00
xy
(M) =
1; C = z
00
yy
(M) = 1  B
2
AC = 1 > 0  ß     ÿ ß
  ß  
z
0
x
= 8 2x
3
x
z
0
y
= 4 4y
3
y
x
4x
2
1
= 0
y
y
2
1
= 0
x x= 0 =
1
2
x =
1
2
y y y= 0 = 1 = 1
¿  ß
ß ¿ ÿ  ß 
M
1
(
0, 0
)
; M
2
(
0, 1
)
; M
3
(
0, 1
)
; M
4
1
2
, 0
; M
5
1
2
, 1
M
6
1
2
, 1
; M
7
1
2
, 0
; M
8
1
2
, 1
; M
9
1
2
, 1
 
z
00
xx
= 24x
2
2; z
00
xy
= 0; z
00
yy
= 12y
2
4.
¿ M
1
(0, 0) A = 2; 4;B = 0; C = B
2
AC = 8 < 0  M
1
 ß ÿ ¿
ß z = 0
¿ M
2
(
0, 1
)
; M
3
(
0, 0;1
)
; A = 2; B = C = 8; B
2
AC = 16 > 0  M M
2
,
3
 ¿  ß ÿ ¿ ß z = 0

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    ß ß ¿
¿ M
4
1
2
, 0
; M
7
1
2
, 0
; A = 4; 4;B = 0; C = B
2
AC = 16 > 0  M
4
, M
7
 ¿  ß ÿ ¿ ß z = 0
¿ M
5
1
2
, 1
; M
6
1
2
, 1
; M
8
1
2
, 1
; M
9
1
2
, 1
; A = 4; 8;B = 0; C = B
2
AC = 32 < 0  M M M M
5
,
6
,
8
,
9
  ß ÿ ß ß  ß ¿  
z =
9
8
  ß  
p
= z
0
x
= 2x + e
(
x
2
+y
2
)
.2 0x =
q
= z
0
y
= 2y + e
(
x
2
+y
2
)
.2y = 0
x = 0
y = 0
¿ M(0, 0)  ß ß ¿  ¿ 
z
00
xx
= 2 + 2.e
(
x
2
+y
2
)
4x
2
.e
(
x
2
+y
2
)
z
00
xy
= 4xy.e
(
x
2
+y
2
)
z
00
yy
= 2 + 2.e
(
x
2
+y
2
)
4y
2
.e
(
x
2
+y
2
)
¿ M(0, 0)  A = 4; 4;B = 0; C = B
2
AC = 16 < 0; A > 0  ¿ M  ß ¿
ÿ ß
 ÿ ß  ß ß
 ¿ ß U R
2
  ß     ÿ ß ÿ  ßf : U R f 
 ¿ x, y ¿   
ϕ(x y, ) = 0
  ¿ ¿ ß (x y
0
,
0
) U ¿  ß ß ϕ(x
0
, y
0
) = 0  f  ÿ ¿ 
ß  ÿ ÿ ß  ß ¿ ß ¿ ß  ¿ V U   f f(x y, ) (x y
0
,
0
 ÿ f f(x y, ) (x y
0
,
0
) ß ß (x y, ) V ¿  ß ß ϕ(x y, ) = 0 ß
( ( (x
0
, y
0
) ÿ ß  ÿ ß  ß ß ÿ  ß f x y, )  ß ß ϕ x y, ) = 0 ÿ
ß  ß ß  ß ÿ   ¿  ß  ¿ ÿ (x y
0
,
0
) ÿ ß ÿ
ϕ(x y y y, ) = 0   ß ÿ  ß y = (x)    (x
0
, (x
0
))  ÿ ß ß 
ÿ  ß ß ¿ ß g(x) = f (x y, (x))  ¿  ß ÿ     ÿ
ß  ß ÿ  ß    ÿ ß ÿ  ÿ  ß ß ¿ ß  
  
 ¿   ÿ ß  ß ß
 z =
1
x
+
1
y
ß ß ß
1
x
2
+
1
y
2
=
1
a
2
 z = x.y ß ß ß x + y = 1

CuuDuongThanCong.com https://fb.com/tailieudientucntt
 ÿ ß ÿ  ß ß ¿ ß
ß ¿  ¿ x =
a
sin
t
; y =
a
cos
t
 
1
x
2
+
1
y
2
=
1
a
2
.  
z
=
1
x
+
1
y
=
sin t
a
+
cos t
a
.
 
z
0
t
=
cos t
a
sin t
a
=
2
a
sin
π
4
t
= 0 t =
π
4
t =
5π
4
ß t =
π
4
  x =
2a a; y =
2  ß ¿ ÿ ß  z

=
2
a
ß t =
5π
4
  x =
2a a; y =
2  ß ¿ ÿ ¿  z

=
2
a
 ÿ ß ß x + y = 1    y = 1 x ¿ z = xy = x x(1 ) ß  ¿
¿  ß x = x x(1 ) ¿ ÿ ¿ ¿ x =
1
2
 z

=
1
4
   ¿     ÿ  ß y = y(x) ÿ ß ß ϕ(x y, ) = 0 
    ÿ ß ß ß  ¿     ÿ ß    ÿ ß
ÿ   ß ÿ       ÿ   ß 
ß   ß ß ¿ ß  ß ¿ ÿ ß ß ß
¿ ÿ
U
 ß ¿
ß 
R
2
f : U R

(x
0
, y
0
)
 ß ÿ ß ÿ 
f
ß ß ß
ϕ(x y, ) = 0
 ÿ ¿ ¿ ¿
  
f (x y x y, ), ϕ( , )
  ¿    ÿ  ß  ¿ ÿ
(x
0
, y
0
)

ϕ
y
(x
0 0
, y ) 6= 0
  ß ¿ ß ß
λ
0
 ß
x
0
, y
0
¿  ß ÿ ß    ß ß
λ, x, y
∂φ
x
= 0
∂φ
y
= 0
∂φ
∂λ
= 0
f
x
(x y, ) + λ
ϕ
x
(x y, ) = 0
f
y
(x y, ) + λ
ϕ
y
(x y, ) = 0
ϕ(x y, ) = 0

ß
φ(x, y, λ) = f (x y, ) + λϕ(x y, )
ÿ ß   
ß     ß ß ¿ ÿ ÿ ß   ß ¿ ß    
¿  ÿ  ß ß ¿ ¿ ÿ M(x
0
, y
0
)  ß ß ß ¿ ÿ ß  ß λ
0


φ ϕ ϕ(x, y, λ
0
) φ(x y
0
,
0
, λ λ
0
) = f (x, y) +
0
(x y, ) f (x
0
, y
0
) λ
0
(x y
0
, y x
0
) = f f(x y, ) (
0
,
0
)
 ¿ M  ß ß ÿ ß ÿ  ß φ(x, y, λ
0
)  M   ß ÿ ß ÿ 
ß f (x y, ) ß ß ß ϕ(x y, ) = 0 ß   M  ¿  ß ÿ ß ÿ  ß
φ(x, y, λ
0
)    ß  ¿ ÿ ÿ ß  ¿     ¿ 
d
2
φ(x y
0
,
0
, λ
0
) =
2
φ
x
2
(x
0
, y
0
, λ
0
)dx
2
+ 2
2
φ
xy
(x
0
, y
0
, λ
0
)dxdy +
2
φ
y
2
(x
0
, y
0
, λ
0
)dy
2

CuuDuongThanCong.com https://fb.com/tailieudientucntt
    ß ß ¿
  dx  dy  ß ß  ß ß ÿ
ϕ
x
(x
0
, y
0
)dx +
ϕ
y
(x
0
, y
0
)dy = 0

dy =
ϕ
x
(x
0
, y
0
)
ϕ
y
(x
0
, y
0
)
dx
 ß ÿ  ÿ dy  d
2
φ(x y
0
,
0
, λ
0
)  
d x y x y
2
φ(
0
,
0
, λ
0
) = G(
0
,
0
, λ
0
)dx
2
ÿ   
¿ G(x y
0
,
0
, λ
0
) > 0  (x
0
, y
0
)  ß ÿ ß  ß ß
¿ G(x y
0
,
0
, λ
0
) < 0  (x
0
, y
0
)  ß ÿ ¿  ß ß
 ¿   ÿ ß  ß ß ÿ  ß z =
1
x
+
1
y
ß ß ß
1
x
2
+
1
y
2
=
1
a
2
ß ¿   ß  φ(x, y, λ) =
1
x
+
1
y
+ λ(
1
x
2
+
1
y
2
1
a
2
) ÿ ß  
∂φ
x
=
1
x
2
2λ
x
3
∂φ
y
=
1
y
2
2λ
y
3
∂φ
∂λ
=
1
x
2
+
1
y
2
1
a
2
= 0
  ÿ  ß ß ¿ 
M
1
(a
2, a
2) ÿ ß λ
1
=
a
2
M
2
(a
2, a
2) ÿ
ß λ
2
=
a
2
 
d
2
φ =
2
φ
x
2
dx
2
+ 2
2
φ
xy
dxdy +
2
φ
y
2
dy
2
=
2
x
3
+
6λ
x
4
dx
2
+
2
y
3
+
6λ
y
4
dy
2
ÿ ß ß
1
x
2
+
1
y
2
1
a
2
= 0  
2
x
3
dx
2
y
3
dy dy= 0  =
y
3
x
3
dx   ß ÿ
d
2
φ  
¿ M
1
d
2
φ(M
1
) =
2
4
a
3
(dx
2
+ dy
2
) =
2
2
4
a
3
(dx
2
) < 0  M
1
 ß ÿ ¿  ß
ß
¿ M
2
d
2
φ(M
2
) =
2
4
a
3
(dx
2
+ dy
2
) =
2
2
4
a
3
(dx
2
) > 0  M
2
 ß ÿ ß  ß
ß

CuuDuongThanCong.com https://fb.com/tailieudientucntt
 ÿ ß ÿ  ß ß ¿ ß
  ß ß ¿  ß ß ¿
¿ ÿ f : A R   ß  ÿ  ¿ ÿ  A ÿ R
2
  f ¿  ß
ß ¿  ß ß ¿  A ß    ß      ß ÿ  ß
¿ ¿ ¿  ß ÿ  ß A   ¿  ß ¿    ß ¿
      ß  ß   ß ÿ    A ÿ A ÿ   ¿
 ÿ ß  ß ß
 ¿    ß ß ¿   ß ß ¿ ÿ   ß
 z = x y y
2
(4 x )     ß ¿ ß  ß x = 0, y = 0, x + y = 6
 z = sin x + sin y + sin(x + y)   ÿ ¿ ß ¿ ß  ß x = 0, x =
π
2
, y = 0, y =
π
2

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Þ  þ þ   Þ
  Þ
 ¿ ¾  
  ß ß
 Þ Þ ¾ Þ      Þ Þ ¾ Þ
 ¿  ¿   ÿ  ¿  ß ¿
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ÿ ÿ                               
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 ß                                           
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       ß ¿ ß                  
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      ¿ ß                  
    ÿ ÿ ÿ                       
   ÿ                            
   ß ÿ  ÿ                         
   ß                                   
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  ¿ ¿                             
  ¿ ÿ    ß                     
  ß ¿   ß                 
       ß                  
ß ß  ¿                                 
 ÿ ÿ ÿ    ß                         
 ß   ¿                           
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 ß  ¿ ß                              
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   ß                                   
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   ß ÿ  ß  ß ¿               
   ß ß ÿ ß ß   ß ÿ             
  ¿ ß ÿ                              
 ¿                                     
    ß ß ¿ ß                       
ß ¿ ÿ  ß ß ¿ ß                          
ß ¿ ÿ  ß ß ¿ ß                      
  ÿ ÿ  ß ß ¿ ß                    
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¿  ÿ  ß ÿ                            
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ÿ ß ÿ  ß ß ¿ ß                           
ÿ ß ÿ                                     CuuDuongThanCong.com
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ÿ ß  ß ß                               
 ß ß ¿   ß ß ¿                       CuuDuongThanCong.com
https://fb.com/tailieudientucntt þ þ  CuuDuongThanCong.com
https://fb.com/tailieudientucntt 
 Þ Þ ¾ Þ 
§  þ Þ  ¾ Þ   ¾ Þ N, Z, Q, R
 ¿   ¿ ÿ ¿ ¿  ¿ ß  ¿  ß ¿ ¿ ÿ
  ¿  ¿    ¿  ß   ¿ ¿ ¿ ¿
 ß ß  ¿ ß ¿   ¿ Q   ß  Z  ¿  ¿ ¿
ÿ ß  ¿ R  ¿ ¿ ÿ ß  ÿ ¿ ¿  ß ¿ ÿ   ß ß ÿ
    ÿ N ⊂ Z ⊂ Q ⊂ R.
§ Þ Þ Þ   ¾
¿ ¿ ß      ¿ 
• |x| ≥ 0, |x| = 0 ⇐⇒x = 0, |x + y| ≤ |x| + |y|
• |x y| ≥ ||x| − |y|| , |x| ≥ A ⇐⇒x A ¿ x ≤−A
• |x| ≤ B ⇐⇒−B x B  CuuDuongThanCong.com
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§ Þ   Þ ¾  Þ ¾  Þ 
  Þ  ¾  ¾  ¾
  þ  þ
 ß   ß
¿ ¿ ß  ß ß    ¿ ¿ ß ¿  ¿ f : X → R  ¿
 ß     X  ß ÿ ÿ f ß ¿ ß ÿ ¿   
  ß   ÿ ß ÿ ¿  ¿  ß ÿ  ß ¿ ß ÿ
¿   ¿ ¿  ß  ß  ß ÿ  ß    ß
¿      ß ¿ ß  ß ß ÿ ¿ 
ß ß   ¿ ¿   ß  ß ¿  ÿ  ß ß 
 ¿ ¿ 
¿  ß ÿ  ß
  ß  ß
  ß ß ¿ ¿  ¿ ß ß ¿
  ¿  ¿  ¿ ÿ ß ß  ¿ ¿ f (x) =  ¿   ¿
  ¿ 
  ß   ÿ    ß ÿ 
 ¿    ÿ ¿    ß T 6= 0(T > 0)   ß 
f (x + T) = f (x)      ß    ß T > 0  ¿
  ÿ ß    ÿ
  ÿ
 ß 
 ß  ß ÿ ß ß ÿ  
 ß  ß ß ß ÿ ß ß ¿  ÿ   ¿
   ß ß    ÿ   ß ÿ  ÿ  ¿ ß ß ÿ
 Þ ß  ß   ¿ y = ax, y = loga x    ÿ ÿ 
  ß  ¿  CuuDuongThanCong.com
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    ß  ¿  ¿
y = xα, y = ax, y = log y y y y a x, = sin x, = cos x, = tg x, = cotg x
y = arcsin x, y = arccos x, y = arctg x, y = arccotg x.
 ß   ß  ¿
  ÿ ß  ß   ¿  ÿ  ÿ ÿ ÿ 
  ¿
 ¿ 
  ÿ  ß  y = 4q 2 1 x+ x lg √ (tan x)  y = arcsin  x  y =
y = arccos(2 sin x) sin πx ß ¿ 
a.  = {π/4 + kπ ≤ x ≤ π/2 + kπ, k ∈ Z} b.  = {−1/3 ≤ x ≤ 1} π π
c.  = {x ≥ 0, x 6∈Z} d.  = {− + kπ ≤ x ≤ 6 + kπ, k ∈ Z} 6
 ¿   ß  ß ÿ  ß  x
y = lg(1 − 2 cos x)  y = arcsin lg ß
¿     {−∞ ≤ 10 y ≤ lg 3}
   {−π/2 ≤ y ≤ π/2}
 ¿   f (x) ¿  1 1 x f x + = x2 +  f = x2. x x2 1 + x x 2 ∀x 6= 1. ß
¿     f (x) = x2 − 2 ß |x|≥2.
  f (x) = 1− x
 ¿    ÿ ÿ  ß  ß   ß   ÿ 1  1 − x  y = 2x + 3.  y = y = 1 + x
(ex + ex) 2 1 ß ¿     y = x − 3 2 2    1 − x
y = y = 1 + x  CuuDuongThanCong.com
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y0 = (ex ex)   ß     ß    ¿    2 ß  ß 1
x ≥ 0, ÿ y = p p
  
(ex + ex)⇒ex = y ± y2 − 1⇒x = ln(y + y2 − 1).  2 [0, +∞) → [1, +∞) 1
x 7→y = (ex + ex) 2 q
ln(y + y2 − 1) ← y
¿  ÿ  ß x ≥ 0  y = ln(x + x2 − 1), x ≥ 1. √
 ß x ≤ 0  ÿ    ÿ  y = ln(x x2 − 1), x ≤ 1.
 ¿    ¿ ¿ ÿ  ß
 f (x) = ax + ax(a > 0) √
 f (x) = ln(x + 1 − x2)
 f (x) = sin x + cos x ß ¿ 
   ß     ß ¿
   ß     ß ¿
   ß    ¿  ¿
 ¿  ÿ  ¿ ¿   ß f (x)   ß  ß ¿ ß
ÿ (−a, a)  ß ß ß ÿ  ¿ ß ¿ ß ÿ ß  ß ¿ 
ß  ß ¿ ß
¿  ß ß f (x) ¿     1 1 f (x) = [ +
f (x) + f (−x)]
[ f (x) − f (−x)] 2 2 g(x) h(x) | {z } | {z }
  g(x)  ß  ß ¿  h(x)  ß  ß ¿
 ¿    ¿     ÿ  ß  ¿ 
 f (x) = A cos λx + B sin λx  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿ ß ¿ ß ß ¿ ÿ ÿ  1 1
f (x) = sin x + sin 2x + sin 3x 2 3
 f (x) = sin2 x
 f (x) = sin(x2) ß ¿ 
 ¿ ÿ T > 0  ß   ÿ  ß    
f (x + T) = f (x)∀x ∈ R
A cos λ(x + T) + B sin λ(x + T) = A cos λx + B sin λx x ∈ R
A[cos λx − cos λ(x + T)] + B[sin λx − sin λ(x + T)] = 0 ∀x ∈ R −λT λT λT ⇔2 sin [A sin(λx + ) + B cos(λx + 2 2 )] = 0 ∀x ∈ R 2 λT ⇔ sin = 0 2 2kπ ⇔ T = . λ
¿  ß   ¿  ß   2π T =  |λ|
      ß sin x ¿  ß   2π  ß sin 2x ¿  ß   1 1
π  ß sin 3x ¿  ß   2π  ¿ f (x) = sin x + sin 2x + sin 3x 3 2 3
¿  ß   T = 2π 
1 − cos 2x ¿  ß  
f (x) = sin2 x = T = π 2
 ¿ ÿ  ß   ¿  ß   T > 0 
sin(x + T)2 = sin(x2)∀x. √
  x = 0⇒T = kπ, k ∈ Z, k > 0. √
  x = π⇒k  ß   ¿ ÿ k = l2, l Z, l > 0 r
   ß  ¿   π 2 x =
¿  ß    ¿ 
 ¿   f (x) = ax + b, f (0) = −2, f (3) = −5.  f (x) 7 ß
¿   f (x) = x − 2. 3
 ¿   f (x) = ax2 + bx + c, f (−2) = 0, f (0) = 1, f (1) = 5.  f (x)  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿ ß 7 ß ¿   17 f (x) = x2 + x + 1. 6 6
 ¿   1
f (x) = (ax + ax), a > 0. ÿ  ¿  2
f (x + y) + f (x y) = 2 f (x) f (y).
 ¿  ¿ ÿ f (x) + f (y) = f (z)  ß z ¿
 f (x) = ax, a 6= 0.
 f (x) = arctan x  1 1 + x f (x) =
 f (x) = lg x 1 − x ß ¿    x + y z = x + y   z = 1 − xy   xy x + y z =   z = x + y 1 + xy
§  Þ
ß   ß   ß ß   ß ß ¿ ß ¿    
  ¿ ß ¿ ß ¿  ¿ ¿  ¿  ß  ¿ 
 ¿ ¿ ß   ß    ß ß  ß ¿  ß
 ß  ß ¿  ß   ß  ÿ   ß ß  ß ß ÿ 
   ¿ ß ¿ ¿    ¿ ß  ß ÿ ß ß ¿
   
  ß ß ß ¿ ∞
   ¿ ß ÿ
  ß ß ¿  ÿ  ¿ ß e
  ¿ ¿
 ß     ¿    ÿ  a  n 1 1 1 a   n = 1 + + + · · · + . 2 3 n  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß
  ¿
 ¿ 
 ß ¿ ÿ   ß  p q p
 xn = n n2 − n  x nπ n =
n(n + a) − n
 xn = n + 3 1 − n3  x
sin2 n − cos3 n n = sin  x 2 2 n = n ß ¿ 
  1   a   0
      0 2 2
 ¿    ß 1 xn = xn−1 + , x x 0 = 1. n−1
 ÿ  ¿  {xn}   ß ¿ ÿ ¿
 ÿ  ¿ lim xn = +∞. n→∞
 ¿   1
un = (1 + )n.ÿ  ¿ {u n
n}  ß  ß   ß ¿ ß
¿   ÿ ¿ ¿ ÿ     1 1 1 + (1 + ) + . . . + (1 + )n. n
) ≥ (n + 1) n+1r n 1 n (1 + 1 1 ⇒(1 + )n+1 ≥ (1 + )n n + 1 n
 ÿ   1 n 1
un = (1 + )n = ∑ Ck n n. nk k=0
k! = 1.2 . . . k ≥ 2k−1∀k ≥ 2 1 1 ⇒ n ) ) Ck
.(n − 1 . . . (n k + 1 < 1 ≤ 1 n. = . k! k! nk nk 2k−1 1 1 1 ⇒un < 1 + 1 + + + . . . + < 3. 2 22 2k−1
 ¿   1 1 sn = 1 + + . . . +
ÿ  ¿ {s 1!
n}   ß ¿ n! ß
¿     lim
un = lim sn = e. n→+∞ n→+∞
 ¿  
1 + a + . . . + an lim
; |a| < 1, |b| < 1.
n→+∞ 1 + b + . . . + bn  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿ ß ß ¿ 
1 + a + . . . + an lim = lim 1 − an+1 1 − b 1 − b . =
n→+∞ 1 + b + . . . + bn n→+∞ 1 − a 1 − bn+1 1 − a
 ¿   lim q p √
2 + 2 + . . . + 2  ¿  n→+∞ q
 ß ¿ ÿ  p { √ = n+1 2 + un un}  ß  ß  ¿
 ß¿ u
  ß ¿ ≤
  ¿  ß ß ¿ { n = 2 + 2 + 0 . . .u+
n ≤ 2  u2 un}  ß
 ß ß ÿ ¿ ÿ lim un = a, 0 < a < 2  ÿ   u2
= 2 + u   n → ∞ n→∞ n+1 n   a2 = a + 2
¿ a = 2  lim q p √ 2 + 2 + . . . + 2 = 2 n→+∞
 ¿   √
lim (n n2 − 1) sin n. n→+∞ ß ¿  √ sin n lim
(n n2 − 1) sin n = lim √
= 0   ¿ ¿ n→+∞
n→+∞ n + n2 − 1
 ¿   lim [cos(ln n) − cos(ln(n + 1))]. n→+∞ ß ¿    ln n + ln(n + 1)
ln n − ln(n + 1)
cos(ln n) − cos(ln(n + 1)) = −2 sin . sin ln n(n + 2 1) ln n n+1 2 = −2 sin sin 2 2  n+1 ln n
0 ≤ |cos(ln n) − cos(ln(n + 1))| ≤ 2 sin ln n 2 ¿  lim sin
n+1 = 0     ß ¿ ¿ n→∞ 2
lim [cos(ln n) − cos(ln(n + 1))] = 0 n→+∞
 ¿  ÿ  ¿ n lim = 0. 2n n→+∞ ß ¿  ⇒ < 2 .
2n = (1 + 1)n > n(n − 1) 0 < n 2 2n n − 1
   ¿   ß ¿ ÿ   CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ¿ ß
 ¿  ÿ  ¿ 2n lim = 0. n→+∞ n! ß ¿    2 2 2 2 2 0 < 2n = . . . . . < 2. n! 1 2 3 nn ≥ 2 n
 ¿    1 1 n lim ( + + . . . + ) n→+∞ 2 22 2n  1 1 n lim ( + + . . . + ) n→+∞ 3 32 3n ß ¿  ÿ  
  Sn − 1 S Sn = 2. 2 nnlim +∞   3 Sn − 1 . S Sn = +∞ 4 3 nnlim
 ¿  ÿ  ¿ lim nnn = 1; lim a = 1, a > 0. n→+∞ n→+∞ ß ¿  ¿ α √ < 2 n = n α2
n − 1⇒n = (1 + α
.  ÿ 
n)n > n(n − 1) 2 n⇒α2n n − 1
 ß ¿ ¿   lim α nn = 0 ¿ lim n→∞ n = 1 n→∞
 ¿ a = 1,   ¿ √
a > 1, 1 ≤ nn
a n n n > a⇒ lim a = 1 n→+∞  ¿ 1
a < 1, ¿ a0 = √ ⇒ n √ lim n a = 1. a a0 = 1⇒ lim n→+∞
 ¿    ¿  ÿ  ¿  ß 1 1 un = 1 + + . . . + 2 n  
 ¿  ÿ  ¿ ¿ a lim a
1 + a2 + . . . an n = a  lim = a. n→+∞ n n→+∞
 ¿  ÿ  ¿ ¿ lim a n
n = a, an > 0∀n  lim n→+∞ a n→+∞
1.a2 . . . an = a.  CuuDuongThanCong.com
https://fb.com/tailieudientucntt 
    Þ ¾ Þ
§   ¾ Þ
   ÿ  ß
    ß         ÿ ÿ  
¿    ÿ  ß ¿   ß ß  ß f (x)   ß ¿  
ß  ß F(x)  ¿  ¿ f (x) ¿ ß ¿   ¿ ¿   ß F(x)  ¿
ß    ß F(x) ÿ ß  ß   ÿ  ß f (x)  ß ¿ D ¿  
F0(x) = f (x) ∀x D
dF(x) = f (x)dx
ß     ¿   ÿ ß  ß  ß  ¿   ¿
¿ ¿ ß      ß  ¿ ÿ ¿ ¿     ÿ  ß 
ß   ¿ F(x)  ß   ÿ  ß f (x)  ¿ D   ß •
F(x) + C   ß   ÿ  ß f (x) ß C  ß ¿ ß ¿ 
ÿ ¿ ß   ÿ  ß •
f (x) ß ¿ ÿ ß ¿ F(x) + C 
 C  ß ¿ ß
 ¿ ß ÿ F(x) + C ß ß ¿ ¿    ÿ  ß f (x) ß ¿
ß C  ÿ   ß    CuuDuongThanCong.com
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ß     ¿ ß ÿ ß  ß f (x)  ß    F(x) + C ß    x D
C  ß   ÿ  ß f (x)  C  ß ¿ ß ¿ 
  ¿ ß ÿ f (x)dx ÿ  ß  Z
 ß ÿ f (x)dx ÿ ß  ß
ÿ ß ¿     ß f (x)dx
f (x) ÿ ß   ß ß ¿   Z ¿
f (x)dx = F(x) + C ß F(x)    ÿ f (x)
  ¿ ÿ   ¿ ß Z 0 Z • f (x)dx
= f (x)  d
f (x)dx = f (x)dx Z Z •
F0(x)dx = F(x) + C 
dF(x) = F(x) + C Z Z •
a f (x)dx = a
f (x)dx a  ¿ ß   Z Z Z
•  [f (x) ±  g  (  x ¿)] dx ß =
  f (x)d x ±   g ¿ (x)dx
¿  ÿ   ¿ ß   ß ¿  Z Z Z
f (x) + βg(x)] dx = α
f (x)dx + β g(x)dx
  α, β   ¿ ß  ß ß ¿ 
  ÿ   ¿  ¿ Z Z xα+1
dx= ln |x| + C xαdx = + C, (α 6= −1) x α + 1 Z Z
sin xdx = − cos x + C
cos xdx = sin x + C Z Z = tgx + C dx c d o x s2 x = −cotgx + C sin2 x Z Z ax axdx =
+ C, (a > 0, a 6= 1)
exdx = ex + C ln a1 a + x 1 x Z = = arctg + C a2 dxx2 Z x2 + d a2x a a ln + C 2a a x x Z √ = arcsin + C dx p Z a2 − x2 adx = ln x + x2 + α + C x2 + α x Z p p arcsin + C 1 a22 a
a2 − x2dx = x a2 − x2 + 2 Z p 1 h p p i x2 + adx =
x x2 + a + a ln
x + x2 + a + C 2  CuuDuongThanCong.com
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       ¿ ß
    ß
ß  ß   ¿   ¿ ÿ ÿ     ÿ ß  ß
      ¿   ÿ    ¿ ß  ß 
  ¿     ß    ÿ   ¿ ¿
 ÿ   ¿ ß Z Z Z
f (x) + βg(x)] dx = α
f (x)dx + β g(x)dx
    ß ß ¿    ß ß ÿ   ß  ¿
  ¿ ÿ   ÿ   ¿ ß ÿ     ¿   5 Z x √ Z Z 5 2 − x3 + C
 ÿ  • 3
(2x x − 3x2)dx = 2 x 2 dx − 3 x2dx = 4 Z Z Z Z = − x 2 cos x + x4 − 4 ln |x| + C
2 sin x + x3 − 1 dx = x3dx dx x 2 sin xdx + Z − 1 + arctgx + C dx Z • 1 x2 = dx = − 1x x2(1 + x2) 1 + x2
   ¿ ß ß ÿ   Z Z
¿  ¿
f (x)dx = F(x) + C 
f (u)du = F(u) + C    u = u(x) 
ß  ß ¿   ÿ   ß ß  ¿ ¿  ¿   ¿  x
ÿ ÿ  ¿   ¿ ß ß ÿ ß ¿   g(x)dx ß ¿
g(x)dx = f (u(x))u0 (x)dx,
  f (x)  ß  ß   ß   ÿ   F(x)   
 ¿  ß  Z Z Z g(x)dx =
f (u(x))u0 (x)dx =
f (u(x))du = F(u(x)) + C Z           
 ß ÿ  ¿ u(x) = ax + b  du = adx   ¿
f (x)dx = F(x) + C Z 1
f (ax + b)dx = F(ax + b) + C a
 ÿ   Z + a cos ax C sin axdx = − 1  Z + a C eaxdx = eax  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß  Z Z
esin x cos xdx = esin x sin
d(sin x) = e x + C  Z Z + 3 tgx + C dx =
(1 + tg2x)d(tgx) = tg3x cos4 x  Z √ Z √ √ 3
x 1 + 3x2dx = 16
1 + 3x2d(1 + 3x2) = 1 1 + 3x2 + C  9 Z Z
arccos x arcsin x π I = √ dx =
− arcsin x arcsin xd(arcsin x) 1 − x2 2  π ⇒ I =
arcsin2 x − 1 arcsin3 x + C 4 3
   ß ¿ Z
   I =
f (x)dx   f (x) ß  ß  ÿ ß   
    ß      ÿ ß  ß  ¿ ß
 ß ¿ x = ϕ(t)   ß ÿ ß ¿   ß ß ¿ t  ß 
ÿ   ß   ¿ 
 ß ¿ ÿ ¿
¿ x = ϕ(t)   ϕ(t)  ß  ß  ß   ¿   ÿ     Z Z I = f (x)dx =
f [φ(t)] φ0(t)dt
¿ ÿ  ß g(t) = f [ϕ(t)] ϕ0(t)      G(t)  t = h(x)  
ß ÿ ÿ  ß x = ϕ(t)   Z
g(t)dt = G(t) + C I = G [h(x)] + C
 ß ¿ ÿ 
¿ t = ψ(x)   ψ(x)  ß  ß  ¿   ÿ   ¿ ÿ 
f (x) = g [ψ(x)] ψ0(x)     Z Z I = f (x)dx =
g [ψ(x)] ψ0(x)dx
¿ ÿ  ß g(t)      ß G(t)  
I = G [ψ(x)] + C
      ¿ ß ¿   ß ¿ ß    ÿ
   ¿ ß ß ¿ ß ¿   ß ÿ ¿ ß   CuuDuongThanCong.com
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 ÿ      Z r dx x 2 − x ¿  I  1  =  ÿ
x = 2 sin2 t, t ∈ 0, π2 r s = tgt x 2 2(s1in2 − stin2 t)
dx = 4 sin t cos tdt, = 2 − x   Z r Z x I dx = 4
sin2 tdt = 2t − sin 2t + C 1 = ß ¿ ¿ 2 − x x ß
   ÿ p 2 t = arcsin x Z r r x x p I1 = dx = 2 arcsin
− 2x x2 + C 2 − x 2
    Z dx e2x ex + 1 ¿ x x I2 =   
e = t e dx = dt Z t Z I2 = dt = 1 − 1
dt = t − ln |t + 1| + C t + 1 t + 1
ß ¿ ¿ x  ÿ I2 = ex − ln(ex + 1) +  C
    Z √ dx I 1 + 4x ¿ 3 =
   ß 
t = 2−x dt = −2−x ln 2dx Z √ = − 1 Z √ = − 1 t ln 2 −d 1 t+ t−2 t d 2 t + 1 p I3 = ln 2
ln(t + t2 + 1) + C ln 2
ß ¿ ¿ x   √
I3 = − 1 ln(2−x + 4−x + 1) + C ln 2
     ÿ ¿
¿ ÿ u = u(x)  v = v(x)    ß  ¿   ÿ   ¿ ¿   Z Z Z
d(uv) = udv + vdu uv = d(uv) = udv + vdu   Z Z udv = uv vdu Z
   I =
f (x)dx  ¿ ß ß
f (x)dx = [g(x)h(x)] dx = g(x) [h(x)dx] = udv  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß Z     ß  ÿ   ÿ    ÿ    ÿ           ÿ       ¿  ß   ß ÿ      ß   ß ¿ u   =  g  (x)  , v  = ÿ h(x ß)dx   
  ß   ln x, ax  ß ÿ   ß ÿ  ÿ ÿ ß Z Z Z •      ß       ß   u =  xnxnekxdx; xn sin kxdx;
xn cos kxdx n    Z
•    
xα lnn xdx α 6= −1  n    ß ß u = lnn x Z Z •      ß    u  =a  rc t  g  kx  xn ¿ ar u ctg = k a x r d csxi;n kxxnd avrcs = in xnkx d d
xxn    ß
 ÿ      ¿ ß  Z Z I1 =
ln xdx = x ln x
dx = x ln x x + C  Z I2  = ¿ x2 sin xdx   ÿ
u = x2, dv = sin xdx v = − cos x Z
I2 = −x2 cos x + 2 x cos xdx ¿   ÿ
u = x, dv = cos xdx v = sin x Z
I2 = −x2 cos x + 2 x sin x − sin xdx
= −x2 cos x + 2x sin x + 2 cos x + C  Z (xexd + x 1)2 I3 =
¿ u = xex; dv = dx v = − 1 ; du = (x + 1)exdx  ÿ (x+1)2 x+1 ex I3 = − xex Z + ex + C = + C xex x + 1 x + 1 + exdx = − x + 1  Z xexdx I 1 + ex 4 =
¿ √1 + ex = t exdx = 2dt    Z 1+ex I4 ß=   2 ¿ [l ¿n
(t − 1) + ln(t + 1)] dt = 2(t
1) ln(t − 1) + 2(t + 1) ln(t + 1) − 4t + C x   Z xexdx √ √ √
= 2(x − 2) 1 + ex + 4 ln 1 + 1 + ex − 2x + C 1 + ex  Z √ x arcsin xdx I 1 − x2 5 = ¿ √
u = arcsin x; dv = xdx √ ⇒ du = dx   ÿ √
; v = − 1 − x2 1−x2 1−x2 p Z p
I5 = − 1 − x2 arcsin x +
dx = − 1 − x2 arcsin x + x + C  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ¿ ß  Z I¿   ÿ 6 = ex cos 2xdx
u = cos 2x; dv = exdx v = ex; du = −2 sin 2xdx Z
I6 = ex cos 2x + 2 ex sin 2xdx ¿   ÿ
u = sin 2x; dv = exdx v = ex; du = 2 cos 2xdx Z
I6 = ex cos 2x + 2 ex sin 2x − 2 ex cos 2xdx
= ex cos 2x + 2ex sin 2x − 4I6 + 5C
¿ I6 = ex ( 5
cos 2x + 2 sin 2x) + C
  ÿ     ¿    ¿ ß ÿ ß ß ¿  
¿   ÿ ÿ ÿ  ÿ   ÿ  ÿ    ß
ß   ¿  ß ß     
     ÿ ÿ ÿ
ß   ß   ÿ ÿ ÿ  ß  ß  ¿ f (x) = P(x)    Q(x)
P(x), Q(x)    ÿ ÿ x ß  ÿ ÿ ÿ  ¿ ÿ  ÿ ß ÿ ß ß 
¿ ÿ  ÿ ß ¿ ß  ß  ÿ ÿ ÿ ÿ ÿ
¿    ÿ  P(x)  Q(x)    ÿ ß   ÿ ÿ ÿ ß ¿ r(x)
f (x) = H(x) + Q(x)
  H(x)   ÿ  r(x)  ¿       r(x)  ß  Q(x)
ÿ ÿ ÿ ÿ ÿ   ÿ  ÿ ÿ  ß  ÿ    ¿
 ¿  ß    ÿ  ÿ ÿ ÿ  ¿ r(x)   ß ÿ ¿ Q(x)
ß ¿ ß ÿ  ÿ   ÿ ¿ ¿ ¿  ÿ ¿   ÿ ß
ÿ ¿ ß ÿ ¿    ÿ ÿ   ß ß ¿ ß ß  ß 
ß ÿ 
  ß ß ¿ ß
¿ ÿ   ß   ß  ÿ ÿ ÿ ÿ ÿ P(x)  ß ß Q(x)
ÿ   ÿ ÿ ÿ ÿ ÿ  ¿ ß   ÿ ¿ ¿ ¿ ¿  ß ¿
    ÿ ß ¿ ß Q(x)   ÿ   ÿ ¿ ¿ ¿ ¿   ß
Q(x) = (x − α 2
1)a1 ...(x − αm)am (x + p1 x + q1)b1 ...(x2 + pn x + qn)bn   α
  ¿ ß
  ß   1 ≤ i m; 1 ≤ j ni, pj, qj ai, bj  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß
• ¿    ÿ Q(x) ¿ ß  ÿ (x − α)aa  ß   
   ÿ  ÿ P(x) ¿ ß  ¿ ÿ ¿ Ai    A  Q(x) (x−α)i i
¿ ß  1 ≤ i a
• ¿    ÿ Q(x) ¿ ß ß ÿ (x2 + px + q)bb  ß 
     ÿ  ÿ P(x) ¿ ß  ¿ ÿ ¿ Bjx+Cj Q(x)
(x2+px+q)j
  B
  ¿ ß  1 ≤ j bj, Cj
  ¿ ÿ   ÿ P(x)    ¿ ß A
¿   ß ¿ Q(x) i, Bj, Cj
ß ß  ¿ ß ß ¿ ß ß ÿ xn, n ∈ R ß  ¿  ¿ ß    ß
ß ¿ ß ¿   ß ß  ß ¿   ÿ ÿ  ¿  Z Z  Adx  Adx
(x a)k x a Z
Z (x2 + px + q)m 
(Mx + N)dx 
(Mx + N)dx
x2 + px + q   Z
= A ln |x a| + C  A x dxa Z = −A + C
(k−1)(xa)k−1  ( A x dx a)k  Z Z
(Mx + N)dx q
Mt + (N Mp/2) = dt
(a = q p2/4, ß ¿ t = x + p/2)
x2 + px + q t2 + a2 Z Z Mtdt (N t M 2 p + / a 2 2 )dt = + t2 + a2 t
= ln(t2 + a2) + (N Mp/2) arctg + C a 2N Mp = 2x + p
ln(x2 + px + q) + p arctg 4q p2 p + C 4q p2  Z Z ( q Mx + N)dx
Mt + (N Mp/2) = dt ( (
a = q p2/4, ß ¿ t = x + p/2)
x2 + px + q)m
(t2 + a2)m Z Z Mtdt (N( − t2 M + pa/22 ) ) m dt = +
(t2 + a2)m Z = − M + C m
2(m−1)(t2+a2)m−1               ÿ    ¿  (t2+a2) Mtdt
 ß       ÿ ¿  ß  ÿ
 ¿ ß  CuuDuongThanCong.com
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 ÿ      ¿ ß  Z dx x4 − x3 (x2 + 2x2 2)( − x 2 − x1 + ) 1 I1  =  
x4 − x3 + 2x2 − 2x + 1 1 A Bx + C = x + = x + +
(x2 + 2)(x − 1)
(x2 + 2)(x − 1) x − 1 x2 + 2
 ß ¿ ß ß  ¿
3 = (A + B)x2 + (C B + 2)x C A + B = 0 A = 1
ß ¿ ß ß ÿ x2, x  ß ß ÿ   ÿ C B + 2 = 0 B = −1 ⇒ C = −1   −C = 1
x4 − x3 + 2x2 − 2x + 3 1 2x = x + − 1 − 1
(x2 + 2)(x − 1) x − 1 2 x2 + 2 x2 + 2
¿   ¿ x2 x I =
+ ln |x − 1| − ln(x2 + 2) − 1 √ arctg 2 2 √ 2 + C 2  Z dx 2x4 + (x 10 + x3 1) + 2( 1 x 7 2 x2 + + 2x 16 + x 3 + ) 5 I2 =  ¿
2x4 + 10x3 + 17x2 + 16x + 5 2 = 2 + − 1 − 4
(x + 1)2(x2 + 2x + 3) x + 1 (x + 1)2 x2 + 2x + 3   1 √
I = 2x + 2 ln |x + 1| + − x + 1 2 2arctg √ + C x + 1 2
    ÿ 
    Z 
ÿ ÿ ß      ß R( sin x,si c n x os , x c  os x)  d
xß   ÿ   ÿ             ß   ¿   ¿ ß        
  ß ß        ÿ   t = tg t  2 2t 2t 2dt sin x = 1 − t2 ; cos x = ; tg x = ; dx = 1 + t2 1 + t2 1 − t2 1 + t2
    ÿ  ß   ÿ  ß ÿ ¿ t  CuuDuongThanCong.com
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 ÿ     Z dx s 1in + xs − in co x s +xc+ os 2x  ¿ Z Z Z
sin x − cos x + 2
d(1 + sin x + cos x) 1 + sindx x + cos x dx = − + 2
1 + sin x + cos x
1 + sin x + cos x
¿ t = tg x   2 Z Z dx
dt = ln |1 + t| + C = 1 + t
1 + sin x + cos x
 ¿ ¿   ÿ
Z sin x − cos x + 2 x
dx = − ln |1 + sin x + cos x| + ln 1 + tg + C
1 + sin x + cos x 2 Z
   ¿
sinm x cosn xdx   m, n   ß 
• ¿ m  ß   ¿  ¿ t = cos x
• ¿ n  ß   ¿  ¿ t = sin x
• ¿ m, n   ß   ¿  ÿ ÿ  ÿ ¿ ¿ 1 − cos 2x 1 + cos 2x sin2 x = ; cos2 x = 2 2 Z
ß  ß   ¿
sink 2x cosl 2xdx
 ÿ      ¿ ß Z • I1 = sin3 x cos2 ¿ xdx  
cos x = t ⇒− sin xdx = dt cos5 x Z Z − t3 + C = − cos3 x + C t55 3 5 3
sin3 x cos2 xdx =
(1 − t2)t2(−dt) = Z • I
ÿ ¿ ¿   2 = sin4 x cos2
ÿ ÿ   xdx 1 + cos 2x 1 Z dx = Z (1 − cos 2x)2 4 2 I 2 3 2 =
1 − cos 2x − cos 2x + cos 2x dx 8 ⇒ I2 = 1 Z 2 dx + 1 Z ¿ x − sin 2x − 1+cos 4x
(1 − sin2 2x)d(sin 2x) 8 2 2 1 sin 2xI ! 2 =
− sin 2x − sin 4x + − sin3 2x x2 2 8 2 + C 8 6  CuuDuongThanCong.com
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ß ß   I   ÿ ÿ  ÿ ¿ ¿ ¿ ÿ ¿    2
ß ¿ ÿ ¿ ¿ ÿ ß ÿ ÿ  ß ¿   ß  ÿ
3 sin x − sin 3x 3 cos x + cos 3x sin3 x = ; cos3 x = 4 4
 ÿ    I    2 1
3 cos 2x + cos 6x I2 = Z + 2
1 − cos 2x − 1 + cos 4x dx 8 1 sin 6x 4 =
− sin 2x − sin 4x + x 2 8 8 + C 8 24 Z
  
R(sin x, cos x)dx  ¿ ¿ ß
• ¿ t = cos x ¿ R(− sin x, cos x) = −R(sin x, cos x)
• ¿ t = sin x ¿ R(sin x, − cos x) = −R(sin x, cos x)
• ¿ t = tg x ¿ R(− sin x, − cos x) = R(sin x, cos x)
 ÿ     Z sin dxxcos4 x ¿   
t = cos x dt = − sin xdx 1 1 Z Z + + − 1 dxdt Z t2 2(t − 1) = 1 t4 = dt sin x cos4 x (1 − t2)t4 1 t − 1 2(t + 1) = − 1 − 1 + 3t3 t ln + C 2 t + 1  1 Z = − 1 − 1 + 1 − cos x + C sin d x x cos4 x 3 cos x3 cos x ln 2 1 + cos x
    ß ÿ  ÿ Z √ Z √
    ¿
R(x, α2 ± x2)dx
R(x, x2 − α2)dx   R(u, v)  
 ß ÿ ÿ Z √
• ¿ x = α tg t ß ß  
R(x, α2 + x2)dx Z √
• ¿ x = α sin t ¿ x = a cos t ß ß  
R(x, α2 − x2)dx α α • ¿ x = ¿ x = Z cos t
ß ß  
R(x, x2 − α2)dx sin t  CuuDuongThanCong.com
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  ß    ÿ  ß ÿ  ÿ  ß ÿ  ß ß 
ß ¿    ¿  Z √ • dx √ = ln x + x2 + α + C x2+α Z √ = arcsin x + C a2−x2 adx √ Z √ +
a2 − x2 + a2 C 2 arcsin xa
a2 − x2dx = 1 2 x Z √ h √ √ i • x2 + adx = 1
x x2 + a + a ln
x + x2 + a 2 + C
 ÿ        Z 2 dx  ( ¿ 1 −  x2)− 3 , π   2 √
dx = cos tdt, 1 − x2 = cos t
x = sin t, t ∈ − π 2 Z Z
= tg t + C = tg(arcsin x) + C c d o t s2 t
(1 − x2)− 32 dx =  Z √
x2 d1x+x2 ¿ x =   
tgt dx = dt cos2 t Z Z = − 1 + C = − 1 + C dx 2 sin t sin(arctgx) √ co s s i t n dt = x2 1 + x2 Z  
+    ¿ m/n r/s R x, ax+b , ..., ax+b dx cx+d cx+d ¿ ax
b = tk ß k  ß  ß ¿ ÿ  ß ß   ß ¿ ÿ ß ß tcx + d
R   ÿ ß  CuuDuongThanCong.com
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§    Þ
 ß     ß
ß  
¿ ÿ  ß f (x)  ß  ß ¿ [a, b]  [a, b]  n
¿ ß [xi, xi+1] ß  ¿ a = x0 < x1 < . . . < xn = b   ß ¿ [xi, xi+1]
 ß ß ξ
  ¿ ß ÿ
i ∈ [xi, xi+1] n−1 S ß n = ∑ 4 −  f
xi = xi+1 xi i ) 4 xi i=0
ß ÿ S ÿ ß  ß   ß
 ¿ ß ¿ ß ¿ ÿ n λ = max 4xi 1≤in
¿ I = lim S  ÿ ß    ¿ n
[a, b]   ÿ ß   λ→0
ß ß ξ  i
I ÿ ß     ß ÿ  ß f (x)  [a, b]   ß  Z
f (x)dx  ß ÿ     ß f (x) ¿   [a, b] b a
   ß       ß f (x)  ¿  [a, b] ÿ 
 ¿ ¿ a < b  ß ¿ b < a  ß Z Z
f (x)dx   a = b b a b
f (x)dx := −
 ß 
f (x)dx = 0 a Z b a
   ¿ ¿ 
ß   ß ß ¿  ÿ ß  ß ß ¿ f (x) ¿   [a, b]  lim(S − λ→0
s) = 0   n+1 n+1 S = ∑ ∑
Mi 4 xi, s = mi 4 xi i=1 i=1 Mi = sup
f (x), mi = inf f (x)
x∈[xi,xi+1] x∈[x x i, i+1]
 ÿ ß      ß ÿ  ÿ  ß  
ß   ¿ f (x)  ÿ  [a, b]  f (x) ¿   [a, b]
ß   ¿ f (x) ß ¿ [a, b]   ß ß ß  ¿  [a, b]  f (x)
¿   [a, b]
ß   ¿ f (x) ß ¿   ß [a, b]  f (x) ¿   [a, b]  CuuDuongThanCong.com
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   ¿ ÿ    ß
Z f (x)dx  b
  ¿ ¿    ¿        ¿ a
ß  f (x) ÿ ¿ ¿  ¿   [a, b] • ¿ b b b
Z [α f (x) + βg(x)]dx = α Z f (x)dx + β Z g(x)dx a a a • ¿
  ¿  [a, b], [a, c], [b, c] ¿ f (x) ¿   ¿  ß  ß ¿
  ¿    ¿  ¿  b c b
Z f (x)dx = Z f (x)dx + Z f (x)dx a a c • ¿
¿ ¿ a < b  
Z b f (x)dx ≥ 0
 ¿ f (x) ≥ 0, ∀x ∈ [a, b]  a Z Z g(x)dx b b  ¿ a
f (x) ≤ g(x)∀x ∈ [a, b] 
f (x)dx
 ¿ f (x) ¿   [a, b] a| f (x)| ¿   [a, b]  b b
| Z f (x)dx |≤ Z | f (x) | dx a a
 ¿ m f (x) ≤ M, f orallx ∈ [a, b]  b
m(b a) ≤ Z f (x)dx M(b a) a  • ¿
ß    ÿ ¿
¿ ÿ f (x) ¿   [a, b]  m f (x) ≤ M, ∀x ∈ [a, b]   ß ¿ µ   b
Z f (x)dx = µ(b a), m ≤ µ ≤ M. a
¿ ß ¿ f (x)  ÿ  [a, b]  ß ¿ c ∈ [a, b]   b
Z f (x)dx = f (c)(b a). a  CuuDuongThanCong.com
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ß    ÿ  ¿ ¿
 f (x)  f (x)g(x) ¿   [a, b]
 m f (x) ≤ M, ∀x ∈ [a, b]
 g(x)  ß ¿  [a, b]   b b
Z f (x)g(x)dx = µ Z g(x)dx, m ≤ µ ≤ M. a a
¿ ß ¿ f (x)  ÿ  [a, b]  ß ¿ c ∈ [a, b]   b b
Z f (x)g(x)dx = f (c) Z g(x)dx. a a
   ß ¿   ß   
¿ ÿ f (x)  ß  ¿   [a, b]   ß ß x ∈ [a, b]  f  ¿  x
 [a, x]   ß  ß F(x) = Z f (t)dta
ß    ¿ f (t) ¿   [a, b]  F(x)  ÿ  [a, b] 
 ¿ f  ÿ ¿    x
F(x)  ¿  ¿ x0 0 ∈ [a, b]
F0(x0) = f (x0)
ß    ÿ  ¿ f (x)  ÿ  ¿  [a, b]
 F(x)  ß   ÿ f (x)  b
Z f (x)dx = F(b) − F(a). a
        ß
 ÿ ÿ  ÿ   ÿ ¿
¿ ÿ u(x), v(x)    ß  ¿   ÿ  [a, b]   b b
Z udv = uv|b − Z vdu a a a  CuuDuongThanCong.com
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 ÿ ÿ   ß ¿ ß b
ß   ß ¿ x := ϕ(t)  I = Z f (x)dx ß f (x)  ÿ  [a, b] a
ÿ ß  ß ¿ x = ϕ(t) ¿   ß ß 
 ϕ(t)  ¿   ÿ  [a, b]
 ϕ(a) = α; ϕ(b) = β.
  t ¿  [α, β] ÿ α ¿ β  x = ϕ(t) ¿   ÿ ÿ a ¿ b
     ÿ b β
Z f (x)dx = Z f [ϕ(t)]ϕ0(t)dt. a α b
ß   ß ¿ t := ϕ(x) ¿ ÿ   ¿   ¿ I = Z 
f [ϕ(x)].ϕ0(x)dx a
  ϕ(x) ¿   ß ¿   ¿   ÿ  [a, b]   b ϕ(b)
Z f [ϕ(x)].ϕ0(x)dx = Z f (t)dt. a ϕ(a)
 ÿ ÿ    ß  ¿
 ß ß  ¿
¿   ¿  ÿ   
     ÿ  x Z 0= f (x)  a f (t)dt x g(x) Z
0 = f (g(x)).g0 (x)  a f (t)dt x
 ÿ     ¿  ß     ÿ  ÿ   ÿ 
ÿ ¿  ÿ  ÿ  CuuDuongThanCong.com
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 ¿    ¿  y y x3 d d d dt a) Z et2 dt b) Z et2dt c) Z dx dy dxx x x2 1 + x4 ß ¿ y x  d
et2dt = − d
et2dt = −ex2(  y  ¿ ß) dx Z dx Z x y y 
d Z et2dt = ey2(  x  ¿ ß) dy x x3 x2 x3  d dt dt d dt 3x2 Z √ = − d Z √ + Z √−2x + dx √ √ 1 + x4 dx 1 + x4 dx = 1 + x8 x2 a a 1 + x4 1 + 12x2
¿   ß ¿ ÿ  ß ÿ   ÿ   ¿  ÿ
  

 ¿   ß ¿ sin x x √ Z tg tdt Z (arctg t)2dt a)A = lim 0 b)B = lim 0 tg xx→0+ √ x→+∞ x2 + 1 Z sin tdt 0 ß ¿ sin x tg x
 ¿  √ lim √ Z tg tdt = lim Z
sin tdt = 0   ÿ  ¿   x→0+ x→0+ 0 0  ! sin x Z 0 √ lim 0 tg tdt p = 1⇒A = 1 x→0+ ! = lim tg(sin x). cos x 0 p cos2 x tg x Z √sin tdt x→0+ sin(tg x). 1 0  CuuDuongThanCong.com
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 ¿  lim √
Z (arctg t)2dt = lim
x2 + 1 = ∞   ÿ  ¿  x→+∞ 0 x→+∞   ! xZ 0 (arctg x)2 π2 π2 lim 0 (arctg t)2dtx = B = x→+∞ √ → √ 4 4 0 =x lim +∞ x2 + 1 x2+1
¿  ÿ ÿ  ÿ ß   ß  ß ¿ ÿ ß ß  ß ¿ ß
¿  ÿ  ÿ  ß    n−1 Sn = ∑ f (ξ ß 4 − i ) 4 xi
xi = xi+1 xi i=0
¿    ¿ [a, b]  n ¿  ß  ¿  ß  ¿ a = x 
0 < x1 < . . . < xn = b   xi = a + (b a) i n b a n−1 Sfn = n
i ) ß ξi ∈ [xi, xi+1] i=0
  ¿  f (x) ¿   [a, b]  ß ξ
 ÿ  ÿ i = xi b
lim b a " ∑ !# Z f (x)dx  n−1 n→∞ i=0 b a f a + .i = a n n
 ¿ ß ξ
 ÿ  ÿ i = xi+1 b
lim b a " ∑ !# Z f (x)dx  n n→∞ i=1 b a f a + .i = a n n
 ¿   ß        ß   ß ¿
 A = lim h nα + 1 nα+β + 1 nα+2β + · · · + 1 i 1 n→∞ nα+(n−1)β  B = lim 1 q q n→∞ p + · · · + 1 + n n 1 + 1 + 1 + 2 n n n ß ¿  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß  ¿ 1 1 1 1 A = lim " + + + · · · + # n→∞ 1α α + β α n + 2β n α + (n−1)β n
 ÿ  ÿ  ß a = 0, b = 1, f (x) = 1  ÿ n α+βx 1 1 1 α + β A = Z dx = α + β ln x β α 0
¿  ÿ  ÿ  ß a = 0, b = 1, f (x) = 1  ÿ α+βx 1 1 α + β A0 = limh + + · · · + ln nα i 1+ β nα + 2β 1β α = A = n→∞ nα + nβ √
  ÿ  ÿ  ß a = 0, b = 1, f (x) = 1 + x  ÿ 1 √ √ B = Z 2 1 + xdx = (2 2 − 1) 0 3 √
¿  ÿ  ÿ  ß a = 0, b = 1, f (x) = 1 + x  ÿ 1 √ B0 = lim r r ! (2 2 − 1) n→∞ 1 n − 1 23 1 + 1 + + · · · + 1 + = B = n n n
 ¿   lim n ! n q 1 (2n)! n→∞ n!
¿      ß  ÿ 
 ¿      e 2 sin2 x cos x
a. Z | ln x | (x + 1)dx d. Z dx (1 + tg2 x)2 0 1e 3
b. Z (x ln x)2dx e. Z r dx x 1 + x 1 0 arcsin π 1 2
c. Z (x3 − 2x + 5)ex 2 dx
f . Z cosn x cos nxdx 0 0 ß ¿  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß
   a, b, c ß ¿ ¿     ÿ ¿  ß   e2 + 5 I 5e3 − 2 a = , I , Ic = 98 − 144 √ 4 b = 27 e  2 sin2 x cos x Id = Z dx (1 + tg2 x)2 0 2 = 4
Z sin2 x. cos x. cos xdx 0 2
= Z sin2 x.(1 − sin2 x)d(sin x) 0 sin3(2) sin7(2) = − 2 sin5(2) + 3 5 7  3 Ie = Z r dx x 1 + x 0 arcsin 3 3 1 1 1 r − Z x. . . dx x (x + 1)2 = x arcsin 0 q 1+x q x+1 1 + x 0 3 1 − x 2 x √ = π − 1 Z x dx 2 x + 1 0 √3 t = π − 1 √ Z 2
.2tdt ¿ x = t) t2 + 1 0 √3 = π − Z 1 − 1 dt t2 + 1 √ 0 3 h i
= π − (t − arctg t) 4π √ = 0 − 3 3  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß  π2
In = Z cosn x cos nxdx 0 π 2 1 = Z cosn xd sin nx n 0 π2 1 π 1 =
2 + Z sin nx.n. cosn−1 x. sin xdx 0 n cosn x sin nx 0 nπ2
= Z sin nx. cosn−1 x. sin xdx 0   π π 2 2
2In = Z cosn x cos nxdx + Z sin nx. cosn−1 x. sin xdx 0 0 π2
= Z cosn−1 x cos(n − 1)xdx 0 = In−1 2n+1
¿     π ß   n In = 1 π 2 2 .I0 = π 2
 ¿   In = Z sinn xdx, Jn = Z cosn xdx 0 0
¿  ÿ   ¿ ÿ  
 ¿  ÿ  ¿ ¿ f (x)  ÿ  [0, 1]  π π 2 2 π π π
a/ Z f (sin x)dx = Z f (sin x)dx,
b/ Z x f (sin x)dx =
Z f (sin x)dx 2 0 0 0 0 ß
¿     ¿ ß   ¿ t = π
x    ¿ t = π − x 2
 ¿   ÿ ¿ ¿ ÿ  ¿   ÿ  π π 2 √ 2 √ π Z sin x √ cos xdx = Z √ √ dx = sin x + cos x 4 0 sin x + cos x 0  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß
 ¿  ¿ ÿ f (x)  ÿ 
  ÿ 
[−a, a](a > 0) a 0
¿ f (x)   ß ¿  [−a, a] a I = Z
2 Z f (x)dx
¿ f (x)   ß ¿  [− − a, a]
a f (x)dx = 0
 ¿   f (x)  ÿ ¿ 
 ÿ  [−a, a] a a f (x)dx Z
= Z f (x)dx ß 1 + bx 0 ≤ b 6= 1 −a 0
 ÿ  π π 1 2 2 1 2x cos 2x I x2 | sin x | 1 = Z dx, I dx, I dx (x2 + Z Z 1)(ex + 1) 2 = 2002x + 2x 3 = 1 + 2x −1 − π2 − π2 b b
 ¿  ÿ  Z xm(a + b x)ndx = Z xn(a + b x)mdx a a 1
 ÿ  I
 ÿ 
n = Z x2(1 − x)ndx 0 n ∑ 1 1 1 (−1)kCk − 2 n. = + k + 3 n + 1 n + 2 n + 3 k=0
¿  ÿ   ¿ ¿ ÿ  
 ¿   f (x), g(x)    ß ¿   [a, b]   f 2(x), g2(x) 
¿   [a, b] ÿ  ¿ ¿ ÿ  (a < b) ! ! ! bZ 2 bZ bZ
a f (x)g(x)dxa
f 2(x)dx . a g2(x)dx
¿ ¿ ÿ  ß
¿    ß ÿ b b
 ¿ Z f2(x)dx = Z g2(x)dx = 0  a a b b
b f2(x) + g2(x) Z
Z | f (x)g(x)|dx ≤ Z dx = 0 2 0 ≤
a f (x)g(x)dxa a
    ¿ ” = ” ¿   CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿  x x x
z0x = y cos ; z0y = 2y sin x cos . y y y 
z0x = y3xy3−1; z0y = 3y2 ln x.xy3  1 ∂ s !
z0x = x2−y2 + 1 x2 − y2 y2 x2+y2 p ∂ = x x2 + y2 x x4 − y4 1 ∂ s !
z0y = x2−y2 + 1 x2 − y2 −y x2+y2 = p ∂y x2 + y2 x4 − y4 
u0x = yzxyz−1; u0y = xyzzyz−1. ln x; u0z = xyzyz ln y ln x  1 − 1 1 2x −2y −2z u0x = e .
x2+y2+z2 .
; u0y = e x2+y2+z2
; u0z = ex2+y2+z2
(x2 + y2 + z2)2
(x2 + y2 + z2)2
(x2 + y2 + z2)2
 ¿  ¿  ÿ  ÿ  ÿ ß ¿  ÿ ÿ  ¿   ÿ 
 ß f (x, y)   x0 arctg y 2   ¿ ¿   x x 6 == 0 0
f (x, y) = x 
x sin y y sin x
¿ (x, y) 6= (0, 0) x2 + y2
f (x, y) = 0
¿ (x, y) = (0, 0) .  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ¿ ß ¿ 
  ß ¿  ß  ÿ ß ß (x, y) 6= (0, y) |x|  2 lim    ÿ   x  = 2  0 R  2 2 x arctg y ≤ π x. arctg y
= 0 = f (0, y) . ¿ f (x, y)  x x→0 x
ß x 6= 0  ¿   ß ¿   ÿ 2x3y z0 , z0y = y 2 x4 + y4 − 2x2y2 x4 + y4 x = arctg x
 ¿ x = 0 f 0
f (h, y) − f (0, y) 0, y = 0 x (0, y) = lim π h→0 2 = arctg h = h y 2 , y 6= 0
f (0, y + k) − f (0, y)
f 0y (0, y) = lim = lim 0 = 0 k k→0 k→0
¿  ¿ f 0x(x, y)  ÿ  R2\ (0, 0)  f 0y (x, y)  ÿ  R2
  ß  ÿ  R2\ (0, 0)  ¿ (0, 0) 
x sin y − ysinx xy − sin x sin y − sin x sin y y y 0 ≤ = ≤ 1 x2 + y2 x2 + y2 x 2 x 
lim x sin y − ysinx x→0 = 0 x2 + y2 ¿ R2 y→0
f (x, y)  ÿ    ¿ ß z  ß  ÿ¿ ÿ  z   =
 yf x  2 ¿ − y
 2 , ß  f   ß ¿  ÿ  ¿ ß ß 1 1 z
z0x + z0y = x y y2 ß ¿    z0 x = y f x2 − y2 .2x, z0 y = f x2 − y2 + y. f x2 − y2 . (−2y)  1 1 z z0x + z0 = x y f
x2y y2 y2 y =
 ¿   ¿  ÿ  ß ÿ   p
 z = eu2−2v2, u = cos x, v = x2 + y2.  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿ . y
 z = ln u2 + v2 , u = xy, v = x
 z = arcsin (x y) , x = 3t, y = 4t3. ß ¿     v0x = x √ ( x = − sin x x2+y2 ; u0u0 v0y = y √ ; x2+y2 y = 0 
z0x = ecos x2−2(x2+y2) [− sin 2x − 4x] .
z0y = ecos x2−2(x2+y2) [−4y] .    ( u0 x = y ( x = 1 y u0 v0v0 ; y = −x y2  y = x 2 z0x = , z0 x 2 y ( y y44 − + 1 1) y =    ( x0t = 3
y0t = 12t2  1 z0t = q 3 − 12t2 1 − (x y)2
 ¿      ¿ ÿ   ß  y x z = sin x2 x + + y2 y .  z = ln tg  z = arctg
 u = xy2z.  x y ß ¿  
dz = cos x2 + y2 (2xdx + 2ydy)  2
dz = sin 2yx xdyydx . . x2  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ¿ 
(x y) dx + (x + y) dy . dz =
(x y)2 + (x + y)2  du = xy2z
y2zdx + 2yz ln xdy + y2 ln xdz . x
 ¿   ¿   √ A = 3 q 1, 03 + 4√ (1, 02)2 + (0, 05)2  B = ln 3 0, 98 − 1 ß ¿ 
   f (x, y) = 3
px2 + y2,∆x = 0,02;∆y = 0,05; x = 1;y = 0.   1 1 f 0x =
2x; f 0y = 2y 3 (x2 + y2)2/3 3 (x2 + y2)2/3   2
f (1 + ∆x, 0 + ∆y) ≈ f (1, 0) + f 0 .0, 02 + 0.0, 05 = 1, 013.
x (1, 0) ∆x + f 0
y (1, 0) ∆y = 1 + 3    √x + 4√ f ( ∆ x, y) = ln 3
y − 1 ; x = 1; y = 1; x = 0, 03; ∆y = 0, 02   1 1 1 1 f 0 x = √ . √ . 3 √ y − 1 2 ; f 0 y = 3 √ y − 1 3 x + 4 3x 3 x + 4 3y 4   1 1
f (1 + ∆x, 1 + ∆y) ≈ f (1, 1) + f 0 .0, 03 +
x (1, 1) ∆x + f 0
y (1, 1) ∆y = 0 + 3 (−0, 02) = 0, 005. 4
 ¿   ¿  ÿ   ß ¿  ß ß      x + y y
x3y y3x = a4;  y0  arctg = ;  y0 a a
 x + y + z = ez;  z0
 x3 + y3 + z3 − 3xyz = 0,  z0 x, z0y x, z0 y ß ¿ 
   ß ¿ F (x, y) = x3y y3x a4 = 0  F0x = 3x2y y3; F0y = x3 − 3y2x. ¿ −F0x y0 =
= − 3x2y y3 F0 y
x3 − 3y2x  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿ 1 a F0 x = = a
   ß ¿ a2+(x+y)2
F (x, y) = arctg x+y y 1+( x+y )2 a  a  F0 y = a
− 1 = a2−a2−(x+y)2 a a a2+(x+y)2
a(a2+(x+y)2) a y0 = . (x + y)2
   ß ¿ F (x, y, z) = x + y + z ez  F0x = 1; F0y = 1; F0z = 1 − ez  −1 −1 z0 ; z0 x = 1 − ez y = 1 − ez
   ß ¿ F (x, y) = x3 + y3 + z3 − 3xyz = 0  F0x = 3x2 − 3yz; F0y = 3y2 − 3xz; F0z =
3z2 − 3xy  3yz − 3x2 3xz − 3y2 z0x = ; z0 3z2 − 3xy
x = 3z2 − 3xy
 ¿   u = x+z  u0
¿ ¿ z   ß ¿ ÿ x, y  ß ß y+z x, u0y
  z.ez = x.ex + y.ey F0
x = − (ex + xex )  F0
y = − (ey + yey) ß
¿    ß F (x, y, z) = zez xex yey = 0 F0z= ez + zez (1 + z0 ez+zez u0
x ) . (y + z) − (x + z) (z0x ) x =
1 + ex+xex − (x + z) ex+xex
ez+zez(y + z)2 = (y + z)2 u0
(x + z) . 1 + z0 − y
(y + z) z0 y
(x + z) . 1 + ey+yey − (y + z) ey+yey ez+zez ez+zez (y + z)2 y = = (y + z)2
 ¿   ¿  ÿ   ß ¿ y(x), z(x)  ß ß ß x + y + z = 0
x2 + y2 + z2 = 1 ß
¿  ¿ ¿   ¿ ÿ    ÿ ß  
1 + y0x + z0x = 0
2x + 2yy0x + 2zz0x = 0  z x
y0x = y z x y z0
x = y z  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ¿
 ¿    z2 + 2 p
¿ x2z0x + 1 z0y = 1z
= y2 − z2  ß  ¿ z = z (x, y) ÿ  x y F0 x = − 2x2 ß
¿    ß F (x, y, z) = z2 + 2 − F0 y √  p y = y2−z2 − x y2 − z2  F0 z = 2z + zy2−z2 2 x2
z0x = 2z + zy2−z2 −yy2−z2
z0y = 2z + zy2−z2 ÿ    z0 y x2z0 = 1 x + y z .
 ¿    ¿   ¿  ÿ   ß   1 z = q yx (x2 + y2)3
 z = x2 ln x2 + y2  z = arctg 3 2x2 + y2 z00 p p 2x =
x2 + y2 + x 2 x2 + y2 p z0 xx = p x x 2 2 + + 2 y2 y2 px2 + y2  z00 p p 2y = ß ¿ 
  x = x z0
x2 + y2 + y 2 x2 + y2 p
y = y x2 + y2 yy = 2xy xy x2 + y2 z00 xy = p = 2 x2 + y2 px2 + y2 2x
2x (x + y) − x2 z00
xx = 2 ln (x + y) + + x2 x + y (x + y)2
z0x = 2x ln (x + y) + x + y  2xx2 x2 z00 xy = + (x + y)2    x + y
z0y = x + y x2 z00
yy = (x + y)2 2xy 1 z00 xx = −yy (x2 + y2)2 z0x = . = 2 x2 x2 + y2  y2 − x2 x 1 1 x z00 − = x2 + y2 + y.2y (x2 + y2)2    1 + y z0y = = (x2 + y2)2 x x2 + y2 xy = −2xy z00 yy = 1 + y 2 (x2 + y2)2 x  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿
 ¿     ¿  ÿ   ß   1
z = xy2 − x2y
 z = 2(x2 + y2) ß ¿ 
   dz = y2 − 2xy dx + 2xy x2 dy 
d2z = −2y (dx)2 + 4 (y x) dxdy+ (2y) (dy)2
   dz = x dx + y 2(x2+y2)2
2(x2+y2)2 dy  y2 − 3x2 x2 − 3y2 d2z = (dx)2 − 4xy dxdy + (dy)2 (x2 + y2)3 (x2 + y2)3 (x2 + y2)3
§ þ Þ þ  Þ Þ ¾ Þ
 ÿ ß ÿ 
ß  
  ß z = f (x, y)  ß  ß ß D  
M0(x0, y0) ∈ D
  ¿  ß f (x, y) ¿ ÿ ß ¿ M ¿ ß ß ß 0
M   ¿   ÿ
M    ß ß
 ¿  ß 0 M0
f (M) − f (M0) ¿
 ß  ¿   ÿ 
ÿ ß  ÿ ß •
f (M) − f (M M0 M0 0 ) > 0
ÿ  ß f ¿ M 0 • ¿
 ß  ¿   ÿ 
ÿ ß  ÿ ¿
f (M) − f (M M0 M0 0 ) < 0
ÿ  ß f ¿ M 0
 ¿ ¿    ÿ ÿ   ß  p = f 0
x (M), q = fy(M), r = fxx”(M), s = fxy ”(M), t = fyy”(M)
ß   ¿  ß f (x, y) ¿ ÿ ß ¿ M  ¿   ¿   p = f 0
x (M), q = fy(M) ß ¿   ¿   ¿ ¿ 
ß   ¿ ÿ  ß z = f (x, y)   ¿   ¿ ¿   ÿ 
ß  ¿   ÿ M  
0(x0, y0) ¿ ÿ ¿ M0
p = q = 0    ¿ 
   ÿ ß ¿ s2 − rt < 0
f (x, y) ¿ ÿ ß ¿ M0
r > 0   ÿ ¿ ¿ r < 0   CuuDuongThanCong.com
https://fb.com/tailieudientucntt ÿ ß ÿ ß ß ¿ ß  ¿   s2 − rt > 0
f (x, y)  ¿ ÿ ß ¿ M0
  ¿ s2 − rt = 0   ¿ ¿ ÿ ß  ß ß M    ß  ÿ ß 0
  ß   ß ÿ   ¿  ß  ß   M  ¿  0
ÿ ß   ¿   ß f (M) − f (M0) ¿   ß ¿  ß 
¿   ÿ M    ÿ ß  ÿ ¿ 0
 ¿   ÿ ß ÿ   ß 
 z = x2 + xy + y2 + x y + 1
 z = x + y x.ey
 z = 2x4 + y4 − x2 − 2y2
 z = x2 + y2 − e−(x2+y2) p = z0
x = 2x + y + 1 = 0 x = −1  ¿   y = 1 ß ¿ 
  ß  q = z0 ⇔
M (−1, 1)  ß ß ¿  ¿ −
y = x + 2y 1 = 0
  A = z00
xx (M) = 2; B = z00
xy(M) = 1; C = z00
yy (M) = 2  B2 − AC = 1 − 4 = −3 <
0. ¿  ß ¿ ÿ ß ¿ M   A > 0  M  ß ÿ ß
  ß   p = 1 − ey = 0 x = 1 y = 0 ⇔
q = 1 − xey = 0
¿  ß  ß ß ¿  ¿ M (1, 0)   A = z00
xx (M) = 0; B = z00 xy(M) =
−1; C = z00yy(M) = −1  B2 − AC = 1 > 0  ß     ÿ ß
  ß   z0
x = 0 ∨ x = 1 ∨
x = 8x3 − 2x x = − 1 2 2 ¿  ß z0 x 4x2 − 1 = 0 ⇔
y = 0 ∨ y = 1 ∨ y = −1
ß ¿ ÿ  ß 
y = 4y3 − 4y y y2 − 1 = 0 1 1 M1 (0, 0) ; M2 (0, 1) ; M3 (0, −1) ; M4 , 0 ; M , 1 2 5 2 1 M6 , −1 ; M , 0 ; M , 1 ; M , −1 2 7 −12 8 −12 9 −12   z00
xx = 24x2 − 2; z00 xy = 0; z00 yy = 12y2 − 4. ¿ M
 ß ÿ ¿
1(0, 0) A = −2; B = 0; C = −4; B2 − AC = −8 < 0  M1 ß z = 0
¿ M2 (0, 1) ; M3 (0, −1) ; A = −2; B = 0; C = 8; B2 − AC = 16 > 0  M2, M3
 ¿  ß ÿ ¿ ß z = 0  CuuDuongThanCong.com
https://fb.com/tailieudientucntt ß ß ¿ ¿   M 1 −1
 ¿  ß ÿ ¿ ß z = 0 4 , 0 ; M
, 0 ; A = 4; B = 0; C = −4; B2 − AC = 16 > 0  M 2 7 2 4, M7 ¿ A C M 1 1 5 = −3 ,2 1< ; M 0  , − M1 ;
  ß ÿ ß ß  ß ¿   5, M M 6, M8, M, 1 ; M B = 0; C = B2 − 2 6 2 8 −129 9 −12, −1 ; A = 4; 8; z = − 9  8 p = z0 x = 0
x = 2x + e−(x2+y2).2x = 0
  ß   q = z0 y = 0 ⇔
¿ M(0, 0)  ß ß ¿ y =  2y  +
¿e−(x2+y2)   .2y = 0 z00
xx = 2 + 2.e−(x2+y2) − 4x2.e−(x2+y2) z00
xy = −4xy.e−(x2+y2) z00
yy = 2 + 2.e−(x2+y2) − 4y2.e−(x2+y2)
¿ M(0, 0)  A = 4; B = 0; C = 4; B2 − AC = −16 < 0; A > 0  ¿ M  ß ¿ ÿ ß
 ÿ ß  ß ß
 ¿ ß U ⊂ R2   ß f : U → R     ÿ ß ÿ  ß f 
 ¿ x, y ¿    ϕ(x, y) = 0
  ¿ ¿ ß (x0, y0) ∈ U ¿  ß ß ϕ(x0, y0) = 0  f  ÿ ¿ ß  ÿ ÿ ß
ß  ¿ ß ¿ ß  ¿ V U   f (x, y) ≤ f (x0, y0
 ÿ f (x, y) ≥ f (x0, y0) ß ß (x, y) ∈ V ¿  ß ß ϕ(x, y) = 0 ß
(x0, y0) ÿ ß  ÿ ß  ß ß ÿ  ß f (x, y)  ß ß ϕ(x, y) = 0 ÿ
ß  ß ß  ß ÿ   ¿  ß  ¿ ÿ (x0, y0) ÿ ß ÿ
ϕ(x, y) = 0   ß ÿ  ß y = y(x)    (x0, y(x0))  ÿ ß ß 
ÿ  ß ß ¿ ß g(x) = f (x, y(x))  ¿  ß ÿ     ÿ
ß  ß ÿ  ß    ÿ ß ÿ  ÿ  ß ß ¿ ß   
  
 ¿   ÿ ß  ß ß
 z = 1 + 1 ß ß ß 1 + 1 = 1 x y x2 y2 a2
 z = x.y ß ß ß x + y = 1  CuuDuongThanCong.com
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 ¿ x = a
; y = a    1 sin t + 1 = 1 .   cos t x2 y2 a2 1 1 sin t cos t z = + = + . x y a a   √ cos t 2 5π z0 − sin t t = ∨ t = a = a π π4 4 √ √ sin
t = 0 ⇔ t = ß √ t = π a 4   
x = 2a; y = 2a  ß ¿ ÿ ß  z 2a 4  = − ß √ √ √ t = 5π   
x = − 2a; y = − 2a  ß ¿ ÿ ¿  z 2 a 4  =
 ÿ ß ß x + y = 1    y = 1 − x ¿ z = xy = x(1 − x) ß  ¿
¿  ß x = x(1 − x) ¿ ÿ ¿ ¿ x = 1  z  2  = 14
   ¿     ÿ  ß y = y(x) ÿ ß ß ϕ(x, y) = 0 
    ÿ ß ß ß  ¿     ÿ ß    ÿ ß
ÿ   ß ÿ       ÿ   ß 
ß   ß ß ¿ ß  ß ¿ ÿ ß ß ß ¿ ÿ U  ß ¿
ß  R2  f : U → R  (x0,y0)  ß ÿ ß ÿ  f ß ß ß ϕ(x,y) = 0
 ÿ ¿ ¿ ¿
   f (x, y), ϕ(x, y)   ¿    ÿ  ß  ¿ ÿ (x0, y0)  ∂ϕ 
∂ (x , y ) 6= 0 y 0 0
  ß ¿ ß ß λ  ß
¿  ß ÿ ß    ß ß 0 x0, y0 λ, x, y  ∂φ ∂ f ∂ = 0 (x, y) = 0 x
∂ (x, y) + λ ∂ϕ xx ∂φ ∂ f  ∂ = 0 (
) + λ ∂ϕ (x, y) = 0 yx, y yy ∂φ ∂λ = 0 ⇔ ϕ(x, y) = 0
ß φ(x, y, λ) = f (x, y) + λϕ(x, y) ÿ ß   
ß     ß ß ¿ ÿ ÿ ß   ß ¿ ß    
¿  ÿ  ß ß ¿ ¿ ÿ M(x  
0, y0)  ß ß ß ¿ ÿ ß  ß λ0  φ(x, y, λ ϕ , ) − f (x ϕ
0) − φ(x0, y0, λ λ
0) = f (x, y) + 0 (x y
0, y0) − λ0 (x0, y0) = f (x, y) − f (x0, y0)
 ¿ M  ß ß ÿ ß ÿ  ß φ(x, y, λ0)  M   ß ÿ ß ÿ 
ß f (x, y) ß ß ß ϕ(x, y) = 0 ß   M  ¿  ß ÿ ß ÿ  ß
φ(x, y, λ0)     ß  ¿ ÿ ÿ ß   ¿     ¿  ∂2φ ∂2φ ∂2φ
d2φ(x0, y0, λ0) = (x (x (x
x2 0, y0, λ0)dx2 + 2 ∂xy 0, y0, λ0)dxdy + ∂y2 0, y0, λ0)dy2  CuuDuongThanCong.com
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  dx  dy  ß ß  ß ß ÿ ∂ϕ ∂ϕ (x (x ∂ 0, y0)dy = 0 x
0, y0)dx + ∂y  ∂ϕ ∂ (x x 0, y0) dy = − dx ∂ϕ ∂ (x y 0, y0)
 ß ÿ  ÿ dy  d2φ(x0, y0, λ0)  
d2φ(x0, y0, λ0) = G(x0, y0, λ0)dx2 ÿ   
• ¿ G(x0, y0, λ0) > 0  (x0, y0)  ß ÿ ß  ß ß
• ¿ G(x0, y0, λ0) < 0  (x0, y0)  ß ÿ ¿  ß ß
 ¿   ÿ ß  ß ß ÿ  ß z = 1 + 1 ß ß ß 1 + 1 = 1 x y x2 y2 a2 ß
¿    ß  φ(x, y, λ) = 1 + 1 + λ( 1 + 1
) ÿ ß   x y x2 − 1 y2 a2 ∂φ ∂ = − 1 − 2λ x x2 x3 ∂φ − 2λ ∂ = − 1 y y2 y3 ∂φ ∂λ = 1 + 1 = 0 x2 − 1 y2 a2 √ √ √ √
  ÿ  ß ß ¿  M
1(a 2, a 2) ÿ ß λ1 = − a M 2, −a 2) ÿ 2 2(−a ß λ    2 = a √2 ∂2φ ∂2φ ∂2φ 6λ 6λ d2φ = dx2 + 2 ∂ dxdy + + + x2 ∂xy 2 x3 2 y3 dy2 = dx2 + dy2 ∂y2 x4 y4
ÿ ß ß 1 + 1 = 0   − 2 = 0  = − y3 x2 − 1 y2 a2
x3 dx − 2y3 dy dy
x3 dx   ß ÿ d2φ   √ √ • ¿
 ß ÿ ¿  ß M  2
2 (dx2) < 0  M 1 d2φ(M1) = −
(dx2 + dy2) = − 24a3 1 4a3 ß √ √ • ¿
 ß ÿ ß  ß M  2
2 (dx2) > 0  M2 2 d2φ(M2) = (dx2 + dy2) = 2 4a3 4a3 ß  CuuDuongThanCong.com
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  ß ß ¿   ß ß ¿
¿ ÿ f : A → R   ß  ÿ  ¿ ÿ  A ÿ R2   f ¿  ß
ß ¿   ß ß ¿  A ß    ß      ß ÿ  ß
¿ ¿ ¿  ß ÿ  ß A   ¿  ß ¿    ß ¿
      ß  ß   ß ÿ    ∂A ÿ A ÿ   ¿
 ÿ ß  ß ß
 ¿    ß ß ¿   ß ß ¿ ÿ   ß
 z = x2y(4 − x y)     ß ¿ ß  ß x = 0, y = 0, x + y = 6
 z = sin x + sin y + sin(x + y)   ÿ ¿ ß ¿ ß  ß x = 0, x =
π , y = 0, y = π  2 2  CuuDuongThanCong.com
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