tai-lieu-bai-giang-1762136874244

Giải tích 1 Chương 6: Nguyên hàm – Tích phân bất định
Nguyê n hà m – Hài phương phà p cơ bà n tì m nguyê n hà m A
TÓM TẮT LÍ THUYẾT I
Định nghĩa – Tính chất 1. Nguyên hàm:
+) F'(x) = f (x) hay dF(x) = f (x)dx  xX
→ F(x) là nguyên hàm của f(x) trong miền X
→F(x) là nguyên hàm f(x) trên X, thì mọi nguyên hàm khác của f(x) trên miền đều có dạng
F(x) + c,c là hằng số tùy ý.
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2. Tích phân bất định:
Kí hiệu I = f (x)dx = F(x) + c 
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Các tích phân cơ bản:
1 Thầy Lam Trường Toán Lý dx 0dx = c  = tgx + c  2 cos x α+1 dx α x x dx =
+ c (α  −1)  = −cot gx + c  α + 1 2 sin x dx dx
= ln x + c (x  0)  = arcsinx + c  x 2 1− x x dx x a a dx = + c  = arctgx + c  ln a 2 1+ x dx = x +c 
chxdx = shx + c  2 x x x
e dx = e + c 
shxdx = chx + c  dx
cos xdx = sin x + c  = thx + c  2 ch x
sin xdx = − cos x + c 
Các tích phân bổ sung: dx 1 x 2 1. = arctg + c  2 2 x 2 2 a x 2 2 − = − + + a + x a a 4. a x dx a x arcsin c  2 2 a dx 2 2 = +  + dx x 2. ln x x a c  = + 2 2  x  a 5. arcsin c 2 2 − a a x dx 1 x − a 3. = ln + c  2 x a 2 2 x − a 2a x + a 2 2 2 2 2 2 6. x  a dx = x  a 
ln x + x + a + c  2 2
3. Các tính chất cơ bản a) ( f (x)dx 
)'= f(x) hay d( f(x)dx  )= f(x)dx
b) F'(x)dx = F(x) + c 
hay dF(x) = F(x) + c  c) c f (x)dx = c f (x)dx   , c là hằng số
2 Thầy Lam Trường Toán Lý
d)  f (x)  g(x) dx = f (x)dx  g(x)dx     
e) Nếu f (x)dx = F(x) + c 
thì f (u)du = F(u) + c 
, với u = u(x) bất kì
II Hai phương pháp tìm nguyên hàm
1. Phương pháp biến đổi số Cho tích phân f
 (x)dx = f φ
  (t) φ' 
(t)dt : Gọi là công thức đổi biến số
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2. Phương pháp tích phân từng phần
Nếu có các hàm số khả vi u = u(x) , v = v(x) : f (x)dx = udv thì ta có công thức: f
 (x)dx = udv = uv− vdu  
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3 Thầy Lam Trường Toán Lý
BÀI TẬP NHẬN BIẾT
Bài 1: Tìm nguyên hàm của các hàm số
a)  tan(2x)dx
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b)  arctan(2x)dx
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ln 2 + x )dx
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4 Thầy Lam Trường Toán Lý d) x  1+ e dx
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Bài 2: Tìm nguyên hàm của các hàm số a)  ( + ) 3x x 1 e dx
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5 Thầy Lam Trường Toán Lý 3 sin x c)  dx cosx
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_______________________________________________________________________________________ dx d)  2 x x + 1
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6 Thầy Lam Trường Toán Lý