积分公式大全
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常 用积分 公式
(一)含有 ax b 的积分 ( a 0 )
1.
dx
=
1 ln
ax
b
C
ax b a
2. ( ax b) dx = 1 ( ax b) 1 C (
1)
a( 1)
3.
x ax
dx = b
1 a2
( ax
b
b ln ax
b)
C
4.
x2 ax
dx b
=
1 a3
1 (ax
2
x2 a2 C
59.
dx = arcsin x C
a2 x2
a
60.
dx
=
x
C
(a2 x2 )3 a 2 a 2 x 2
61.
x dx= a2 x2
a2 x2 C
成功是必须的
62.
x
dx = 1
C
ax b
9.
dx = 1 x(ax b)2 b( ax b)
1 ax b b2 ln x
C
(二)含有 ax b 的积分
10.
ax bdx = 2 (ax b)3 C 3a
11. x ax
bdx
=
2 15a
2
(3ax
2b) ( ax
b)3
C
12. x2 ax bdx = 2 3 (15a2x2 12abx 8b2 ) ( ax b)3 C 105a
a4 ln( x
8
x2 a 2 ) C
43.
x2
a2 dx =
x2
a2
a ln
x2
a2
a
C
x
x
44.
x2 a2 x2 dx =
x2 a2 ln( x
x
x2 a2 ) C
(七)含有 x2 a2 (a 0) 的积分
45.
dx = x arch x
x2 a2 x
a
C1 = ln x
x2 a2 C
46.
dx
=
1
(x2
a 2)3
C
3
x2 a2 C
56. x 2 x 2 a 2 dx = x (2 x 2 a 2) x 2 a 2 8
a4 ln x
8
x2 a2 C
57.
x2 a2 dx=
x2
a2
a a arccos
C
x
x
58.
x2 a2 x2 dx =
x2 a2 ln x
x
(八)含有 a2 x2 (a 0) 的积分
dx = x
x2
a2
a2 ln( x
x2 a2
2
2
x2 a 2 ) C
成功是必须的
36.
x2
dx =
(x2 a2)3
x ln( x
x2 a2
x2 a2 ) C
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37.
dx
= 1 ln x2 a2 a C
x x2 a2 a
x
38.
dx
=
x2 x2 a2
x2 a2
a2x
C
39.
x2 a2dx = x x2
13.
x ax
b
dx =
2 3a 2
( ax
2b) ax
b
C
成功是必须的
14.
x2 ax
dx = b
2 15a 3
(3a2
x2
4abx
8b2 ) ax
b
C
15.
d源自文库 =
x ax b
1 ln ax b b C ( b 0) b ax b b
2
ax b
arctan
C (b 0)
b
b
16.
dx =
x2 ax b
=
x
C
(x2 a2)3
a2 x2 a2
47.
x dx = x2 a2 C x2 a2
48.
x
dx =
(x2 a 2)3
1
C
x2 a2
成功是必须的
49. 50.
x2
dx = x x2
a2
a2 ln x
x2 a2
2
2
x2
dx =
(x2 a2)3
x ln x
x2 a2
x2 a2 C x2 a2 C
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b)
1 dx 2b ax2 b
(五)含有 ax2 bx c ( a 0) 的积分
29.
dx ax2 bx
=
c
2 arctan 2ax b C
4ac b2
4ac b2
(b2 4ac)
1
2ax b
ln
b2 4ac 2ax b
b2 4ac C (b2 4ac) b2 4ac
x 30. ax2 bx
dx = 1 ln ax2 c 2a
51.
dx = 1 arccos a C
x x2 a2 a
x
52.
dx
=
x2 x2 a2
x2 a2 a2x
C
53.
x2
a 2dx = x x2
a2
a2 ln x
2
2
x2 a2 C
54.
( x2 a 2 )3 dx= x (2 x2 5a 2) x2 a 2 3 a 4 ln x
8
8
55.
2
xx
a
2
dx
(b 0) C (b 0)
成功是必须的
25.
dx x(ax 2
= 1 ln b) 2b
x2 ax 2
b
C
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26.
dx x 2(ax2
=
b)
1 bx
a dx b ax2 b
27.
dx x 3( ax2
a ax 2 b b) = 2b2 ln x2
1 2bx2
C
28.
dx =
x
(ax 2 b) 2 2b( ax2
bx
c
b
dx
2a ax 2 bx
c
(六)含有 x2 a2 (a 0) 的积分
31.
dx
x
x2
= arsh
a2
a
C1= ln( x
x2 a2 ) C
32.
dx
=
x
C
( x2 a2 )3 a 2 x2 a2
33.
x
dx = x2 a2 C
x2 a2
34.
x
dx =
(x2 a2)3
1
C
x2 a2
35.
x2
ax b a
dx
bx 2b x ax b
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17. ax b dx = 2 ax b b
dx
x
x ax b
18.
ax x2
b dx =
ax b a dx
x
2 x ax b
(三)含有 x2 a 2 的积分
dx 1
x
19.
x2
a2
= arctan
a
a
C
dx
x
2n 3
dx
20. ( x2 a 2) n = 2( n 1)a 2 ( x 2 a2 )n 1 2(n 1)a 2 ( x2 a 2) n 1
dx 1 x a 21. x2 a2 = 2a ln x a C
(四)含有 ax2 b(a 0) 的积分
1 arctan a x C
dx
ab
b
22.
ax 2
=
b
1
ax b
ln
2 ab ax b
23.
x
2
dx = 1 ln ax2
b
C
ax b 2a
24.
x2 ax 2
dx = x ba
b a
dx ax2 b
2
b)
2 b(ax
b)
2
b ln ax
b
C
5.
dx
=
1 ax ln
b
C
x( ax b) b
x
6.
dx x2 (ax
=
b)
1 bx
a ax b b2 ln x
C
7.
x ( ax
b) 2dx =
1 a2 (ln
ax
b
b )C
ax b
8.
x2 (ax b) 2dx
=
1 a3
( ax
b
2b ln ax
b
b2 )C
a2
a2 ln( x
2
2
x2 a2 ) C
40.
( x2 a2 )3dx = x (2 x2 5a 2) x2 a 2 3 a4 ln( x
8
8
41. x x2 a 2dx = 1 ( x 2 a 2)3 C 3
x2 a2 ) C
42. x 2 x 2
a2 dx = x (2 x 2 a 2 ) x 2 a 2 8
常 用积分 公式
(一)含有 ax b 的积分 ( a 0 )
1.
dx
=
1 ln
ax
b
C
ax b a
2. ( ax b) dx = 1 ( ax b) 1 C (
1)
a( 1)
3.
x ax
dx = b
1 a2
( ax
b
b ln ax
b)
C
4.
x2 ax
dx b
=
1 a3
1 (ax
2
x2 a2 C
59.
dx = arcsin x C
a2 x2
a
60.
dx
=
x
C
(a2 x2 )3 a 2 a 2 x 2
61.
x dx= a2 x2
a2 x2 C
成功是必须的
62.
x
dx = 1
C
ax b
9.
dx = 1 x(ax b)2 b( ax b)
1 ax b b2 ln x
C
(二)含有 ax b 的积分
10.
ax bdx = 2 (ax b)3 C 3a
11. x ax
bdx
=
2 15a
2
(3ax
2b) ( ax
b)3
C
12. x2 ax bdx = 2 3 (15a2x2 12abx 8b2 ) ( ax b)3 C 105a
a4 ln( x
8
x2 a 2 ) C
43.
x2
a2 dx =
x2
a2
a ln
x2
a2
a
C
x
x
44.
x2 a2 x2 dx =
x2 a2 ln( x
x
x2 a2 ) C
(七)含有 x2 a2 (a 0) 的积分
45.
dx = x arch x
x2 a2 x
a
C1 = ln x
x2 a2 C
46.
dx
=
1
(x2
a 2)3
C
3
x2 a2 C
56. x 2 x 2 a 2 dx = x (2 x 2 a 2) x 2 a 2 8
a4 ln x
8
x2 a2 C
57.
x2 a2 dx=
x2
a2
a a arccos
C
x
x
58.
x2 a2 x2 dx =
x2 a2 ln x
x
(八)含有 a2 x2 (a 0) 的积分
dx = x
x2
a2
a2 ln( x
x2 a2
2
2
x2 a 2 ) C
成功是必须的
36.
x2
dx =
(x2 a2)3
x ln( x
x2 a2
x2 a2 ) C
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37.
dx
= 1 ln x2 a2 a C
x x2 a2 a
x
38.
dx
=
x2 x2 a2
x2 a2
a2x
C
39.
x2 a2dx = x x2
13.
x ax
b
dx =
2 3a 2
( ax
2b) ax
b
C
成功是必须的
14.
x2 ax
dx = b
2 15a 3
(3a2
x2
4abx
8b2 ) ax
b
C
15.
d源自文库 =
x ax b
1 ln ax b b C ( b 0) b ax b b
2
ax b
arctan
C (b 0)
b
b
16.
dx =
x2 ax b
=
x
C
(x2 a2)3
a2 x2 a2
47.
x dx = x2 a2 C x2 a2
48.
x
dx =
(x2 a 2)3
1
C
x2 a2
成功是必须的
49. 50.
x2
dx = x x2
a2
a2 ln x
x2 a2
2
2
x2
dx =
(x2 a2)3
x ln x
x2 a2
x2 a2 C x2 a2 C
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b)
1 dx 2b ax2 b
(五)含有 ax2 bx c ( a 0) 的积分
29.
dx ax2 bx
=
c
2 arctan 2ax b C
4ac b2
4ac b2
(b2 4ac)
1
2ax b
ln
b2 4ac 2ax b
b2 4ac C (b2 4ac) b2 4ac
x 30. ax2 bx
dx = 1 ln ax2 c 2a
51.
dx = 1 arccos a C
x x2 a2 a
x
52.
dx
=
x2 x2 a2
x2 a2 a2x
C
53.
x2
a 2dx = x x2
a2
a2 ln x
2
2
x2 a2 C
54.
( x2 a 2 )3 dx= x (2 x2 5a 2) x2 a 2 3 a 4 ln x
8
8
55.
2
xx
a
2
dx
(b 0) C (b 0)
成功是必须的
25.
dx x(ax 2
= 1 ln b) 2b
x2 ax 2
b
C
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26.
dx x 2(ax2
=
b)
1 bx
a dx b ax2 b
27.
dx x 3( ax2
a ax 2 b b) = 2b2 ln x2
1 2bx2
C
28.
dx =
x
(ax 2 b) 2 2b( ax2
bx
c
b
dx
2a ax 2 bx
c
(六)含有 x2 a2 (a 0) 的积分
31.
dx
x
x2
= arsh
a2
a
C1= ln( x
x2 a2 ) C
32.
dx
=
x
C
( x2 a2 )3 a 2 x2 a2
33.
x
dx = x2 a2 C
x2 a2
34.
x
dx =
(x2 a2)3
1
C
x2 a2
35.
x2
ax b a
dx
bx 2b x ax b
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17. ax b dx = 2 ax b b
dx
x
x ax b
18.
ax x2
b dx =
ax b a dx
x
2 x ax b
(三)含有 x2 a 2 的积分
dx 1
x
19.
x2
a2
= arctan
a
a
C
dx
x
2n 3
dx
20. ( x2 a 2) n = 2( n 1)a 2 ( x 2 a2 )n 1 2(n 1)a 2 ( x2 a 2) n 1
dx 1 x a 21. x2 a2 = 2a ln x a C
(四)含有 ax2 b(a 0) 的积分
1 arctan a x C
dx
ab
b
22.
ax 2
=
b
1
ax b
ln
2 ab ax b
23.
x
2
dx = 1 ln ax2
b
C
ax b 2a
24.
x2 ax 2
dx = x ba
b a
dx ax2 b
2
b)
2 b(ax
b)
2
b ln ax
b
C
5.
dx
=
1 ax ln
b
C
x( ax b) b
x
6.
dx x2 (ax
=
b)
1 bx
a ax b b2 ln x
C
7.
x ( ax
b) 2dx =
1 a2 (ln
ax
b
b )C
ax b
8.
x2 (ax b) 2dx
=
1 a3
( ax
b
2b ln ax
b
b2 )C
a2
a2 ln( x
2
2
x2 a2 ) C
40.
( x2 a2 )3dx = x (2 x2 5a 2) x2 a 2 3 a4 ln( x
8
8
41. x x2 a 2dx = 1 ( x 2 a 2)3 C 3
x2 a2 ) C
42. x 2 x 2
a2 dx = x (2 x 2 a 2 ) x 2 a 2 8