integral-table 积分表
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(52)
x ln(ax + b) dx =
x2 −
ln(ax + b)
(53)
1 1 x ln a2 − b2 x2 dx = − x2 + 2 2
x2 −
ln a2 − b2 x2
(54)
(ln x)2 dx = 2x − 2x ln x + x(ln x)2
(55)
(ln x)3 dx = −6x + x(ln x)3 − 3x(ln x)2 + 6x ln x
x − 2x a
(50)
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
x+a − 2x x−a
(51) ln ax2 + bx + c dx =
1√ 2ax + b b −2x+ + x ln ax2 + bx + c 4ac − b2 tan−1 √ 2 a 2a 4ac − b bx 1 2 1 − x + 2a 4 2 b2 a2 a2 b2
(33)
√
a2
(34)
√
√ x dx = x2 ± a2 x 2 ± a2
(35)
√
√ x dx = − a2 − x2 a2 − x 2
(36)
√
√ 1 √ x2 1 dx = x x2 ± a2 ∓ a2 ln x + x2 ± a2 2 2 x 2 ± a2
(37) √
ax2
b + 2ax √ 2 4ac − b2 ln 2ax + b + 2 + bx + c dx = ax + bx + c+ 4a 8a3/2
or or
(21)
√
ax + b dx =
2b 2x + 3a 3
√ ax + b
(22)
(ax + b)3/2 dx =
2 (ax + b)5/2 5a
(23)
√ x 2 √ dx = (x ∓ 2a) x ± a 3 x±a x dx = − x(a − x) − a tan−1 a−x x dx = a+x √ x ax + b dx = x(a − x) x−a
√ 1 √ 1 x a2 − x2 dx = x a2 − x2 + a2 tan−1 √ 2 2 a2 − x 2 √ 1 2 x ± a2 x x2 ± a2 dx = 3
(31)
3/2
(32)
√
√ 1 dx = ln x + x2 ± a2 x 2 ± a2 1 x dx = sin−1 2 a −x
(84) (85) sin2 ax cos2 bxdx =
cos2 ax sin ax dx = −
x sin 2ax sin[2(a − b)x] sin 2bx sin[2(a + b)x] − − + − 4 8a 16(a − b) 8b 16(a + b) x sin 4ax − 8 32a
a(ax + b)
b b2 x + b) dx = − 2 + 12a 8a x 3 √
x3 (ax
√ b3 + b)+ 5/2 ln a x + 8a
a(ax + b)
(29)
√ 1 √ 1 x2 ± a2 dx = x x2 ± a2 ± a2 ln x + x2 ± a2 2 2 3
(30)
(44)
(45)
xn ln x dx = xn+1
,
n = −1
(46)
1 ln ax dx = (ln ax)2 x 2 ln x 1 ln x dx = − − 2 x x x b a
(47)
(48)
ln(ax + b) dx =
x+ 5
ln(ax + b) − x, a = 0
(49)
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
(75)
cos ax dx =
(76)
cos2 ax dx =
x sin 2ax + 2 4a
(77)
cos3 axdx =
3 sin ax sin 3ax + 4a 12a 8
(78)
cosp axdx = −
1+p 1 3+p 1 cos1+p ax × 2 F1 , , , cos2 ax a(1 + p) 2 2 2 1 2 1 1 sin x + c1 = − cos2 x + c2 = − cos 2x + c3 2 2 4 cos[(a − b)x] cos[(a + b)x] − ,a = b 2(a − b) 2(a + b) sin[(2a − b)x] sin bx sin[(2a + b)x] + − 4(2a − b) 2b 4(2a + b) 1 3 sin x 3
(63)
xe
2 ax
dx =
x2 2x 2 − 2 + 3 a a a
eax
(64)
x3 ex dx = x3 − 3x2 + 6x − 6 ex
(65)
xn eax dx =
xn eax n − a a
xn−1 eax dx
(66)
x e
n ax
(−1)n dx = n+1 Γ[1 + n, −ax], where Γ(a, x) = a √ √ i π dx = − √ erf ix a 2 a 7
(12)
(13)
ax2
(14)
1 a+x 1 dx = ln , a=b (x + a)(x + b) b−a b+x a x dx = + ln |a + x| (x + a)2 a+x x 1 b 2ax + b dx = ln |ax2 + bx + c|− √ tan−1 √ 2 + bx + c 2a a 4ac − b 4ac − b2
(5) 1 1 dx = − 2 (x + a) x+a (x + a)n+1 + c, n = −1 n+1 (x + a)n+1 ((n + 1)x − a) (n + 1)(n + 2)
(6)
(x + a)n dx =
(7)
x(x + a)n dx =
(8)
1 dx = tan−1 x 1 + x2 1 1 −1 x dx = tan a2 + x 2 a a 1
(71) 1 sin ax dx = − cos ax a x sin 2ax − 2 4a
(72)
sin2 ax dx =
(73)
sin3 ax dx = −
3 cos ax cos 3ax + 4a 12a
(74)
1 1−n 3 1 sinn ax dx = − cos ax 2 F1 , , , cos2 ax a 2 2 2 1 sin ax a
Table of Integrals
Basic Forms
(1) xn dx = 1 xn+1 , n = −1 n+1 1 dx = ln |x| x
(2)
(3)
udv = uv −
vdu
(4)
1 1 dx = ln |ax + b| ax + b a
Integrals of Rational Functions
(79)
cos x sin x dx =
(80)
cos ax sin bx dx =
(81)
sin2 ax cos bx dx = −
(82)
sin2 x cos x dx =
(83)
cos2 ax sin bx dx =
cos[(2a − b)x] cos bx cos[(2a + b)x] − − 4(2a − b) 2b 4(2a + b) 1 cos3 ax 3a
1 csc2 ax dx = − cot ax a 1 1 csc3 x dx = − cot x csc x + ln | csc x − cot x| 2 2 1 cscn x cot x dx = − cscn x, n = 0 n
(15)
(16)
ax2
Integrals with Roots
(17) √ 2 x − a dx = (x − a)3/2 3 √ 1 dx = 2 x ± a x±a
(18)
√
(19)
√
√ 1 dx = −2 a − x a−x 2
(20)
x x − a dx =
√
2a (x − a)3/2 + 2 (x − a)5/2 , 3 5 4 2 x(x − a)3/2 − 15 (x − a)5/2 , 3 2 (2a + 3x)(x − a)3/2 15
(94)
sec x tan x dx = sec x
(95)
sec2 x tan x dx =
1 sec2 x 2
(96)
secn x tan x dx =
