Integrals of Rational Functions
Table of Integrals
Basic Forms
Z
(1)
xn dx =
Z
(2)
Z
(3)
1
dx = ln |x|
x
udv = uv −
Z
vdu
1
1
dx = ln |ax + b|
ax + b
a
Z
(4)
1
xn+1 , n 6= −1
n+1
Integrals of Rational Functions
(5)
Z
(6)
(7)
Z
(x + a)n dx =
x(x + a)n dx =
Z
(8)
(9)
1
1
dx = −
2
(x + a)
x+a
Z
Z
(x + a)n+1
+ c, n 6= −1
n+1
(x + a)n+1 ((n + 1)x − a)
(n + 1)(n + 2)
1
dx = tan−1 x
1 + x2
1
1
−1 x
dx
=
tan
a2 + x2
a
a
1
x
1
dx = ln |a2 + x2 |
2
+x
2
(10)
Z
a2
(11)
Z
x2
−1 x
dx
=
x
−
a
tan
a2 + x2
a
(12)
Z
(13)
ax2
Z
(14)
2
2ax + b
1
dx = √
tan−1 √
+ bx + c
4ac − b2
4ac − b2
1
1
a+x
dx =
ln
, a 6= b
(x + a)(x + b)
b−a b+x
Z
(15)
(16)
1
1
x3
dx = x2 − a2 ln |a2 + x2 |
2
2
a +x
2
2
Z
Z
ax2
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
Integrals with Roots
(17)
(18)
(19)
Z
Z
Z
√
2
x − a dx = (x − a)3/2
3
√
√
√
1
dx = 2 x ± a
x±a
√
1
dx = −2 a − x
a−x
2
(20)
Z
√
x x − a dx =
Z √
(21)
2a
(x − a)3/2 + 52 (x − a)5/2 ,
3
4
2
x(x − a)3/2 − 15
(x − a)5/2 ,
3
2
(2a + 3x)(x − a)3/2
15
ax + b dx =
2b 2x
+
3a
3
(22)
Z
(ax + b)3/2 dx =
(23)
Z
√
(24)
Z r
2
(ax + b)5/2
5a
√
2
x
dx = (x ∓ 2a) x ± a
3
x±a
√
x ax + b dx =
(26)
(27)
Z p
x(ax + b) dx =
(29)
ax + b
p
x(a − x)
x−a
p
√
√
x
dx = x(a + x) − a ln x + x + a
a+x
Z
(28)
Z p
√
p
x
dx = − x(a − x) − a tan−1
a−x
Z r
(25)
or
or
√
2
2
2 2
ax + b
(−2b
+
abx
+
3a
x
)
15a2
√
i
p
p
1 h
2
(2ax
+
b)
ax(ax
+
b)
−
b
ln
x
+
a(ax
+
b)
a
3/2
4a
√
2
p
p
b
x
b3
b
3
3
− 2 +
x (ax + b) dx =
x (ax + b)+ 5/2 ln a x + a(ax + b)
12a 8a x 3
8a
Z √
√
1 √
1
x2 ± a2 dx = x x2 ± a2 ± a2 ln x + x2 ± a2
2
2
3
Z √
(30)
x
1 √
1
a2 − x2 dx = x a2 − x2 + a2 tan−1 √
2
2
a2 − x2
Z
(31)
Z
(32)
√
3/2
1 2
x ± a2
x x2 ± a2 dx =
3
√
Z
(33)
Z
(34)
Z
(35)
Z
(36)
(37)
Z √
√
1
dx = ln x + x2 ± a2
x 2 ± a2
ax2
√
√
√
√
a2
x
1
dx = sin−1
2
a
−x
√
x
dx = x2 ± a2
x2 ± a2
√
x
dx = − a2 − x2
a2 − x2
√
x2
1 √
1
dx = x x2 ± a2 ∓ a2 ln x + x2 ± a2
2
2
x 2 ± a2
p
b + 2ax √ 2
4ac − b2
2
+
ln 2ax + b + 2 a(ax + bx c)
+ bx + c dx =
ax + bx + c+
3/2
4a
8a
(38)
Z √
1 √ √ 2
2
2
a
ax
+
bx
+
c
−3b
+
2abx
+
8a(c
+
ax
)
x ax2 + bx + c dx =
2
48a5/2
√ √
+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + x
4
(39)
(40)
Z
√
Z
p
1
1
2
√
dx = √ ln 2ax + b + 2 a(ax + bx + c)
2
a
ax + bx + c
p
x
1√ 2
b
2
dx =
ax + bx + c− 3/2 ln 2ax + b + 2 a(ax + bx + c)
2
a
2a
ax + bx + c
Z
(41)
x
dx
√
=
(a2 + x2 )3/2
a2 a2 + x2
Integrals with Logarithms
Z
(42)
ln ax dx = x ln ax − x
(43)
Z
x2
1
x ln x dx = x2 ln x −
2
4
(44)
Z
1
x3
x2 ln x dx = x3 ln x −
3
9
(45)
Z
n
x ln x dx = x
n+1
1
ln x
−
n + 1 (n + 1)2
(46)
Z
ln ax
1
dx = (ln ax)2
x
2
(47)
Z
1 ln x
ln x
dx = − −
2
x
x
x
(48)
Z
ln(ax + b) dx =
b
x+
a
5
,
n 6= −1
ln(ax + b) − x, a 6= 0
(49)
Z
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
(50)
Z
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
x
− 2x
a
x+a
− 2x
x−a
(51)
Z
2ax
+
b
1√
b
−1
2
ln ax + bx + c dx =
4ac − b2 tan √
−2x+
+ x ln ax2 + bx + c
a
2a
4ac − b2
(52)
Z
(53)
Z
(56)
(57)
ln(ax + b)
a2
x − 2
b
ln a2 − b2 x2
1
1
dx = − x2 +
2
2
2
2 2
x ln a − b x
Z
(54)
(55)
b2
x − 2
a
bx 1 2 1
− x +
x ln(ax + b) dx =
2a 4
2
Z
2
2
(ln x)2 dx = 2x − 2x ln x + x(ln x)2
(ln x)3 dx = −6x + x(ln x)3 − 3x(ln x)2 + 6x ln x
Z
Z
x(ln x)2 dx =
x2 1 2
1
+ x (ln x)2 − x2 ln x
4
2
2
x2 (ln x)2 dx =
2x3 1 3
2
+ x (ln x)2 − x3 ln x
27
3
9
6
Integrals with Exponentials
Z
(58)
(59)
Z
√
xe
ax
√
Z x
√
1 √ ax
i π
2
2
dx =
xe + 3/2 erf i ax , where erf(x) = √
e−t dt
a
2a
π 0
Z
(60)
Z
(61)
Z
(62)
(63)
Z
(64)
Z
Z
(65)
(66)
(67)
Z
1
eax dx = eax
a
n ax
x e
xex dx = (x − 1)ex
xe
ax
dx =
1
x
− 2
a a
eax
x2 ex dx = x2 − 2x + 2 ex
2 ax
xe
dx =
x2 2x
2
− 2 + 3
a
a
a
eax
x3 ex dx = x3 − 3x2 + 6x − 6 ex
n ax
x e
xn eax n
−
dx =
a
a
Z
xn−1 eax dx
(−1)n
dx = n+1 Γ[1 + n, −ax], where Γ(a, x) =
a
Z
e
ax2
√
√
i π
dx = − √ erf ix a
2 a
7
Z
∞
ta−1 e−t dt
x
Z
(68)
e
−ax2
Z
(69)
Z
(70)
√
√
π
dx = √ erf x a
2 a
2
xe−ax dx = −
1
dx =
4
2 −ax2
xe
r
1 −ax2
e
2a
√
x
π
2
erf(x a) − e−ax
3
a
2a
Integrals with Trigonometric Functions
Z
(71)
Z
(72)
Z
(73)
(74)
Z
sin2 ax dx =
sin3 ax dx = −
x sin 2ax
−
2
4a
3 cos ax cos 3ax
+
4a
12a
1
1 1−n 3
2
sin ax dx = − cos ax 2 F1 ,
, , cos ax
a
2
2
2
n
Z
(75)
Z
(76)
(77)
1
sin ax dx = − cos ax
a
Z
cos ax dx =
cos2 ax dx =
cos3 axdx =
1
sin ax
a
x sin 2ax
+
2
4a
3 sin ax sin 3ax
+
4a
12a
8
(78)
Z
1
1+p 1 3+p
1+p
2
cos axdx = −
cos ax × 2 F1
, ,
, cos ax
a(1 + p)
2 2 2
(79)
Z
cos x sin x dx =
p
Z
(80)
(81)
Z
cos ax sin bx dx =
(82)
(83)
(84)
cos[(a − b)x] cos[(a + b)x]
−
, a 6= b
2(a − b)
2(a + b)
sin2 ax cos bx dx = −
Z
Z
1
1
1 2
sin x + c1 = − cos2 x + c2 = − cos 2x + c3
2
2
4
sin2 x cos x dx =
cos2 ax sin bx dx =
Z
sin[(2a − b)x] sin bx sin[(2a + b)x]
+
−
4(2a − b)
2b
4(2a + b)
1 3
sin x
3
cos[(2a − b)x] cos bx cos[(2a + b)x]
−
−
4(2a − b)
2b
4(2a + b)
cos2 ax sin ax dx = −
1
cos3 ax
3a
(85)
Z
x sin 2ax sin[2(a − b)x] sin 2bx sin[2(a + b)x]
sin2 ax cos2 bxdx = −
−
+
−
4
8a
16(a − b)
8b
16(a + b)
(86)
(87)
Z
sin2 ax cos2 ax dx =
Z
x sin 4ax
−
8
32a
1
tan ax dx = − ln cos ax
a
9
Z
(88)
(89)
Z
tann+1 ax
× 2 F1
tan ax dx =
a(1 + n)
n
Z
(90)
(91)
tan2 ax dx = −x +
Z
tan3 axdx =
Z
Z
sec3 x dx =
Z
(94)
Z
(95)
Z
(96)
(97)
n+1
n+3
, 1,
, − tan2 ax
2
2
1
1
ln cos ax +
sec2 ax
a
2a
x
sec x dx = ln | sec x + tan x| = 2 tanh−1 tan
2
(92)
(93)
1
tan ax
a
Z
sec2 ax dx =
1
tan ax
a
1
1
sec x tan x + ln | sec x + tan x|
2
2
sec x tan x dx = sec x
sec2 x tan x dx =
secn x tan x dx =
1
sec2 x
2
1
secn x, n 6= 0
n
x
csc x dx = ln tan = ln | csc x − cot x| + C
2
10
Z
(98)
Z
(99)
1
csc2 ax dx = − cot ax
a
1
1
csc3 x dx = − cot x csc x + ln | csc x − cot x|
2
2
Z
(100)
1
cscn x cot x dx = − cscn x, n 6= 0
n
Z
(101)
sec x csc x dx = ln | tan x|
Products of Trigonometric Functions and Monomials
Z
(102)
Z
(103)
Z
(104)
Z
(105)
Z
(106)
(107)
Z
x cos x dx = cos x + x sin x
x cos ax dx =
x
1
cos ax + sin ax
2
a
a
x2 cos x dx = 2x cos x + x2 − 2 sin x
x2 cos ax dx =
2x cos ax a2 x2 − 2
+
sin ax
a2
a3
1
xn cos xdx = − (i)n+1 [Γ(n + 1, −ix) + (−1)n Γ(n + 1, ix)]
2
1
xn cos ax dx = (ia)1−n [(−1)n Γ(n + 1, −iax) − Γ(n + 1, ixa)]
2
11
Z
(108)
Z
(109)
Z
(110)
Z
(111)
(112)
Z
x sin x dx = −x cos x + sin x
x sin ax dx = −
x cos ax sin ax
+
a
a2
x2 sin x dx = 2 − x2 cos x + 2x sin x
x2 sin ax dx =
2 − a2 x 2
2x sin ax
cos ax +
3
a
a2
1
xn sin x dx = − (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)]
2
(113)
Z
x cos2 x dx =
1
x2 1
+ cos 2x + x sin 2x
4
8
4
(114)
Z
x sin2 x dx =
x2 1
1
− cos 2x − x sin 2x
4
8
4
(115)
Z
x tan2 x dx = −
(116)
Z
x2
+ ln cos x + x tan x
2
x sec2 x dx = ln cos x + x tan x
12
Products of Trigonometric Functions and Exponentials
Z
(117)
Z
(118)
ebx sin ax dx =
Z
(119)
Z
(120)
1
ex sin x dx = ex (sin x − cos x)
2
a2
1
ebx (b sin ax − a cos ax)
+ b2
1
ex cos x dx = ex (sin x + cos x)
2
ebx cos ax dx =
a2
1
ebx (a sin ax + b cos ax)
2
+b
(121)
Z
1
xex sin x dx = ex (cos x − x cos x + x sin x)
2
(122)
Z
1
xex cos x dx = ex (x cos x − sin x + x sin x)
2
Integrals of Hyperbolic Functions
Z
(123)
(124)
(125)
Z
cosh ax dx =
1
sinh ax
a
ax
e
2
[a cosh bx − b sinh bx] a 6= b
2
eax cosh bx dx = a2ax− b
e + x
a=b
4a
2
Z
sinh ax dx =
13
1
cosh ax
a
Z
(126)
(127)
ax
e
2
[−b cosh bx + a sinh bx] a 6= b
2
ax
e sinh bx dx = a2ax− b
e − x
a=b
4a
2
Z
1
tanh ax dx = ln cosh ax
a
(a+2b)x
h
i
e
a
a
2bx
1
+
F
,
1,
2
+
,
−e
2
1
Z
2b
(a + 2b)
h 2b a
i
a
1
ax
ax
2bx
(128)
e tanh bx dx =
a 6= b
− e 2 F1 1, , 1 + , −e
a −1 ax 2b
2b
ax
e − 2 tan [e ]
a=b
a
Z
1
[a sin ax cosh bx + b cos ax sinh bx]
(129)
cos ax cosh bx dx = 2
a + b2
Z
(130)
(131)
Z
sin ax sinh bx dx =
Z
(133)
(134)
sin ax cosh bx dx =
Z
(132)
cos ax sinh bx dx =
Z
a2
a2
1
[b cos ax cosh bx + a sin ax sinh bx]
+ b2
1
[−a cos ax cosh bx + b sin ax sinh bx]
+ b2
a2
1
[b cosh bx sin ax − a cos ax sinh bx]
+ b2
sinh ax cosh axdx =
sinh ax cosh bx dx =
b2
1
[−2ax + sinh 2ax]
4a
1
[b cosh bx sinh ax − a cosh ax sinh bx]
− a2
c 2012. From http://integral-table.com, last revised September 8, 2012. This mate
rial is provided as is without warranty or representation about the accuracy, correctness or
suitability of this material for any purpose. This work is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License. To view
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14
Basic Forms
Z
(1)
xn dx =
Z
(2)
Z
(3)
1
dx = ln |x|
x
udv = uv −
Z
vdu
1
1
dx = ln |ax + b|
ax + b
a
Z
(4)
1
xn+1 , n 6= −1
n+1
Integrals of Rational Functions
(5)
Z
(6)
(7)
Z
(x + a)n dx =
x(x + a)n dx =
Z
(8)
(9)
1
1
dx = −
2
(x + a)
x+a
Z
Z
(x + a)n+1
+ c, n 6= −1
n+1
(x + a)n+1 ((n + 1)x − a)
(n + 1)(n + 2)
1
dx = tan−1 x
1 + x2
1
1
−1 x
dx
=
tan
a2 + x2
a
a
1
x
1
dx = ln |a2 + x2 |
2
+x
2
(10)
Z
a2
(11)
Z
x2
−1 x
dx
=
x
−
a
tan
a2 + x2
a
(12)
Z
(13)
ax2
Z
(14)
2
2ax + b
1
dx = √
tan−1 √
+ bx + c
4ac − b2
4ac − b2
1
1
a+x
dx =
ln
, a 6= b
(x + a)(x + b)
b−a b+x
Z
(15)
(16)
1
1
x3
dx = x2 − a2 ln |a2 + x2 |
2
2
a +x
2
2
Z
Z
ax2
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
Integrals with Roots
(17)
(18)
(19)
Z
Z
Z
√
2
x − a dx = (x − a)3/2
3
√
√
√
1
dx = 2 x ± a
x±a
√
1
dx = −2 a − x
a−x
2
(20)
Z
√
x x − a dx =
Z √
(21)
2a
(x − a)3/2 + 52 (x − a)5/2 ,
3
4
2
x(x − a)3/2 − 15
(x − a)5/2 ,
3
2
(2a + 3x)(x − a)3/2
15
ax + b dx =
2b 2x
+
3a
3
(22)
Z
(ax + b)3/2 dx =
(23)
Z
√
(24)
Z r
2
(ax + b)5/2
5a
√
2
x
dx = (x ∓ 2a) x ± a
3
x±a
√
x ax + b dx =
(26)
(27)
Z p
x(ax + b) dx =
(29)
ax + b
p
x(a − x)
x−a
p
√
√
x
dx = x(a + x) − a ln x + x + a
a+x
Z
(28)
Z p
√
p
x
dx = − x(a − x) − a tan−1
a−x
Z r
(25)
or
or
√
2
2
2 2
ax + b
(−2b
+
abx
+
3a
x
)
15a2
√
i
p
p
1 h
2
(2ax
+
b)
ax(ax
+
b)
−
b
ln
x
+
a(ax
+
b)
a
3/2
4a
√
2
p
p
b
x
b3
b
3
3
− 2 +
x (ax + b) dx =
x (ax + b)+ 5/2 ln a x + a(ax + b)
12a 8a x 3
8a
Z √
√
1 √
1
x2 ± a2 dx = x x2 ± a2 ± a2 ln x + x2 ± a2
2
2
3
Z √
(30)
x
1 √
1
a2 − x2 dx = x a2 − x2 + a2 tan−1 √
2
2
a2 − x2
Z
(31)
Z
(32)
√
3/2
1 2
x ± a2
x x2 ± a2 dx =
3
√
Z
(33)
Z
(34)
Z
(35)
Z
(36)
(37)
Z √
√
1
dx = ln x + x2 ± a2
x 2 ± a2
ax2
√
√
√
√
a2
x
1
dx = sin−1
2
a
−x
√
x
dx = x2 ± a2
x2 ± a2
√
x
dx = − a2 − x2
a2 − x2
√
x2
1 √
1
dx = x x2 ± a2 ∓ a2 ln x + x2 ± a2
2
2
x 2 ± a2
p
b + 2ax √ 2
4ac − b2
2
+
ln 2ax + b + 2 a(ax + bx c)
+ bx + c dx =
ax + bx + c+
3/2
4a
8a
(38)
Z √
1 √ √ 2
2
2
a
ax
+
bx
+
c
−3b
+
2abx
+
8a(c
+
ax
)
x ax2 + bx + c dx =
2
48a5/2
√ √
+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + x
4
(39)
(40)
Z
√
Z
p
1
1
2
√
dx = √ ln 2ax + b + 2 a(ax + bx + c)
2
a
ax + bx + c
p
x
1√ 2
b
2
dx =
ax + bx + c− 3/2 ln 2ax + b + 2 a(ax + bx + c)
2
a
2a
ax + bx + c
Z
(41)
x
dx
√
=
(a2 + x2 )3/2
a2 a2 + x2
Integrals with Logarithms
Z
(42)
ln ax dx = x ln ax − x
(43)
Z
x2
1
x ln x dx = x2 ln x −
2
4
(44)
Z
1
x3
x2 ln x dx = x3 ln x −
3
9
(45)
Z
n
x ln x dx = x
n+1
1
ln x
−
n + 1 (n + 1)2
(46)
Z
ln ax
1
dx = (ln ax)2
x
2
(47)
Z
1 ln x
ln x
dx = − −
2
x
x
x
(48)
Z
ln(ax + b) dx =
b
x+
a
5
,
n 6= −1
ln(ax + b) − x, a 6= 0
(49)
Z
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
(50)
Z
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
x
− 2x
a
x+a
− 2x
x−a
(51)
Z
2ax
+
b
1√
b
−1
2
ln ax + bx + c dx =
4ac − b2 tan √
−2x+
+ x ln ax2 + bx + c
a
2a
4ac − b2
(52)
Z
(53)
Z
(56)
(57)
ln(ax + b)
a2
x − 2
b
ln a2 − b2 x2
1
1
dx = − x2 +
2
2
2
2 2
x ln a − b x
Z
(54)
(55)
b2
x − 2
a
bx 1 2 1
− x +
x ln(ax + b) dx =
2a 4
2
Z
2
2
(ln x)2 dx = 2x − 2x ln x + x(ln x)2
(ln x)3 dx = −6x + x(ln x)3 − 3x(ln x)2 + 6x ln x
Z
Z
x(ln x)2 dx =
x2 1 2
1
+ x (ln x)2 − x2 ln x
4
2
2
x2 (ln x)2 dx =
2x3 1 3
2
+ x (ln x)2 − x3 ln x
27
3
9
6
Integrals with Exponentials
Z
(58)
(59)
Z
√
xe
ax
√
Z x
√
1 √ ax
i π
2
2
dx =
xe + 3/2 erf i ax , where erf(x) = √
e−t dt
a
2a
π 0
Z
(60)
Z
(61)
Z
(62)
(63)
Z
(64)
Z
Z
(65)
(66)
(67)
Z
1
eax dx = eax
a
n ax
x e
xex dx = (x − 1)ex
xe
ax
dx =
1
x
− 2
a a
eax
x2 ex dx = x2 − 2x + 2 ex
2 ax
xe
dx =
x2 2x
2
− 2 + 3
a
a
a
eax
x3 ex dx = x3 − 3x2 + 6x − 6 ex
n ax
x e
xn eax n
−
dx =
a
a
Z
xn−1 eax dx
(−1)n
dx = n+1 Γ[1 + n, −ax], where Γ(a, x) =
a
Z
e
ax2
√
√
i π
dx = − √ erf ix a
2 a
7
Z
∞
ta−1 e−t dt
x
Z
(68)
e
−ax2
Z
(69)
Z
(70)
√
√
π
dx = √ erf x a
2 a
2
xe−ax dx = −
1
dx =
4
2 −ax2
xe
r
1 −ax2
e
2a
√
x
π
2
erf(x a) − e−ax
3
a
2a
Integrals with Trigonometric Functions
Z
(71)
Z
(72)
Z
(73)
(74)
Z
sin2 ax dx =
sin3 ax dx = −
x sin 2ax
−
2
4a
3 cos ax cos 3ax
+
4a
12a
1
1 1−n 3
2
sin ax dx = − cos ax 2 F1 ,
, , cos ax
a
2
2
2
n
Z
(75)
Z
(76)
(77)
1
sin ax dx = − cos ax
a
Z
cos ax dx =
cos2 ax dx =
cos3 axdx =
1
sin ax
a
x sin 2ax
+
2
4a
3 sin ax sin 3ax
+
4a
12a
8
(78)
Z
1
1+p 1 3+p
1+p
2
cos axdx = −
cos ax × 2 F1
, ,
, cos ax
a(1 + p)
2 2 2
(79)
Z
cos x sin x dx =
p
Z
(80)
(81)
Z
cos ax sin bx dx =
(82)
(83)
(84)
cos[(a − b)x] cos[(a + b)x]
−
, a 6= b
2(a − b)
2(a + b)
sin2 ax cos bx dx = −
Z
Z
1
1
1 2
sin x + c1 = − cos2 x + c2 = − cos 2x + c3
2
2
4
sin2 x cos x dx =
cos2 ax sin bx dx =
Z
sin[(2a − b)x] sin bx sin[(2a + b)x]
+
−
4(2a − b)
2b
4(2a + b)
1 3
sin x
3
cos[(2a − b)x] cos bx cos[(2a + b)x]
−
−
4(2a − b)
2b
4(2a + b)
cos2 ax sin ax dx = −
1
cos3 ax
3a
(85)
Z
x sin 2ax sin[2(a − b)x] sin 2bx sin[2(a + b)x]
sin2 ax cos2 bxdx = −
−
+
−
4
8a
16(a − b)
8b
16(a + b)
(86)
(87)
Z
sin2 ax cos2 ax dx =
Z
x sin 4ax
−
8
32a
1
tan ax dx = − ln cos ax
a
9
Z
(88)
(89)
Z
tann+1 ax
× 2 F1
tan ax dx =
a(1 + n)
n
Z
(90)
(91)
tan2 ax dx = −x +
Z
tan3 axdx =
Z
Z
sec3 x dx =
Z
(94)
Z
(95)
Z
(96)
(97)
n+1
n+3
, 1,
, − tan2 ax
2
2
1
1
ln cos ax +
sec2 ax
a
2a
x
sec x dx = ln | sec x + tan x| = 2 tanh−1 tan
2
(92)
(93)
1
tan ax
a
Z
sec2 ax dx =
1
tan ax
a
1
1
sec x tan x + ln | sec x + tan x|
2
2
sec x tan x dx = sec x
sec2 x tan x dx =
secn x tan x dx =
1
sec2 x
2
1
secn x, n 6= 0
n
x
csc x dx = ln tan = ln | csc x − cot x| + C
2
10
Z
(98)
Z
(99)
1
csc2 ax dx = − cot ax
a
1
1
csc3 x dx = − cot x csc x + ln | csc x − cot x|
2
2
Z
(100)
1
cscn x cot x dx = − cscn x, n 6= 0
n
Z
(101)
sec x csc x dx = ln | tan x|
Products of Trigonometric Functions and Monomials
Z
(102)
Z
(103)
Z
(104)
Z
(105)
Z
(106)
(107)
Z
x cos x dx = cos x + x sin x
x cos ax dx =
x
1
cos ax + sin ax
2
a
a
x2 cos x dx = 2x cos x + x2 − 2 sin x
x2 cos ax dx =
2x cos ax a2 x2 − 2
+
sin ax
a2
a3
1
xn cos xdx = − (i)n+1 [Γ(n + 1, −ix) + (−1)n Γ(n + 1, ix)]
2
1
xn cos ax dx = (ia)1−n [(−1)n Γ(n + 1, −iax) − Γ(n + 1, ixa)]
2
11
Z
(108)
Z
(109)
Z
(110)
Z
(111)
(112)
Z
x sin x dx = −x cos x + sin x
x sin ax dx = −
x cos ax sin ax
+
a
a2
x2 sin x dx = 2 − x2 cos x + 2x sin x
x2 sin ax dx =
2 − a2 x 2
2x sin ax
cos ax +
3
a
a2
1
xn sin x dx = − (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)]
2
(113)
Z
x cos2 x dx =
1
x2 1
+ cos 2x + x sin 2x
4
8
4
(114)
Z
x sin2 x dx =
x2 1
1
− cos 2x − x sin 2x
4
8
4
(115)
Z
x tan2 x dx = −
(116)
Z
x2
+ ln cos x + x tan x
2
x sec2 x dx = ln cos x + x tan x
12
Products of Trigonometric Functions and Exponentials
Z
(117)
Z
(118)
ebx sin ax dx =
Z
(119)
Z
(120)
1
ex sin x dx = ex (sin x − cos x)
2
a2
1
ebx (b sin ax − a cos ax)
+ b2
1
ex cos x dx = ex (sin x + cos x)
2
ebx cos ax dx =
a2
1
ebx (a sin ax + b cos ax)
2
+b
(121)
Z
1
xex sin x dx = ex (cos x − x cos x + x sin x)
2
(122)
Z
1
xex cos x dx = ex (x cos x − sin x + x sin x)
2
Integrals of Hyperbolic Functions
Z
(123)
(124)
(125)
Z
cosh ax dx =
1
sinh ax
a
ax
e
2
[a cosh bx − b sinh bx] a 6= b
2
eax cosh bx dx = a2ax− b
e + x
a=b
4a
2
Z
sinh ax dx =
13
1
cosh ax
a
Z
(126)
(127)
ax
e
2
[−b cosh bx + a sinh bx] a 6= b
2
ax
e sinh bx dx = a2ax− b
e − x
a=b
4a
2
Z
1
tanh ax dx = ln cosh ax
a
(a+2b)x
h
i
e
a
a
2bx
1
+
F
,
1,
2
+
,
−e
2
1
Z
2b
(a + 2b)
h 2b a
i
a
1
ax
ax
2bx
(128)
e tanh bx dx =
a 6= b
− e 2 F1 1, , 1 + , −e
a −1 ax 2b
2b
ax
e − 2 tan [e ]
a=b
a
Z
1
[a sin ax cosh bx + b cos ax sinh bx]
(129)
cos ax cosh bx dx = 2
a + b2
Z
(130)
(131)
Z
sin ax sinh bx dx =
Z
(133)
(134)
sin ax cosh bx dx =
Z
(132)
cos ax sinh bx dx =
Z
a2
a2
1
[b cos ax cosh bx + a sin ax sinh bx]
+ b2
1
[−a cos ax cosh bx + b sin ax sinh bx]
+ b2
a2
1
[b cosh bx sin ax − a cos ax sinh bx]
+ b2
sinh ax cosh axdx =
sinh ax cosh bx dx =
b2
1
[−2ax + sinh 2ax]
4a
1
[b cosh bx sinh ax − a cosh ax sinh bx]
− a2
c 2012. From http://integral-table.com, last revised September 8, 2012. This mate
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suitability of this material for any purpose. This work is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License. To view
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