single-page-integral-table

Table of Integrals∗
Basic Forms
Integrals with Logarithms
√
x ax + bdx =
Z
Z
1
x dx =
xn+1
n+1
Z
1
dx = ln |x|
x
Z
Z
udv = uv − vdu
Z
n
(1)
Z p
x(ax + b)dx =
p
1
(2ax + b) ax(ax + b)
4a3/2
i
p
√
(27)
−b2 ln a x + a(ax + b)
(3)
Z
Z p
Z
x(x + a)n dx =
Z
Z
Z
Z
(x + a)n+1 ((n + 1)x − a)
(n + 1)(n + 2)
1
1
x
dx = tan−1
a2 + x2
a
a
(9)
Z p
x3
1
1
dx = x2 − a2 ln |a2 + x2 |
2
a + x2
2
2
(12)
Z
Z
√
1
a−x
√
dx = −2 a − x
√
2
2
x x − adx = a(x − a)3/2 + (x − a)5/2
3
5
Z
√
Z
Z
Z
Z r
1
√
x±a
√
dx = 2 x ± a
ax + bdx =
2b
2x
+
3a
3
√
ax + b
a2 − x2 dx =
p
3/2
1 2
x x2 ± a2 dx =
x ± a2
3
1
√
x2 ± a2
Z
Z
∗
x
a+x
p
dx = ln x + x2 ± a2
bx
1
− x2
2a
4
1
b2
+
x2 − 2 ln(ax + b)
2
a
(32)
(33)
p
x
√
dx = x2 ± a2
2
2
x ±a
(34)
p
x
√
dx = − a2 − x2
2
2
a −x
(35)
1
x ln a2 − b2 x2 dx = − x2 +
2
1
a2
2
x − 2 ln a2 − b2 x2
2
b
(18)
Z
(19)
Z
Z
√
Z
Z
1 √ p 2
2 a ax + bx + c
48a5/2
× −3b2 + 2abx + 8a(c + ax2 )
√ p
+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + c
(38)
Z
(24)
p
1
1
√
dx = √ ln 2ax + b + 2 a(ax2 + bx + c)
2
a
ax + bx + c
(39)
1p 2
x
√
dx =
ax + bx + c
a
ax2 + bx + c
p
b
− 3/2 ln 2ax + b + 2 a(ax2 + bx + c)
2a
(25)
dx
x
= √
(a2 + x2 )3/2
a2 a2 + x2
1 ax
e
a
(50)
xex dx = (x − 1)ex
xeax dx =
1
x
− 2
a
a
(51)
(52)
eax
(53)
x2 ex dx = x2 − 2x + 2 ex
(54)
(37)
p
x ax2 + bx + c =
Z
eax dx =
√
√ 1 √ ax
i π
xe + 3/2 erf i ax ,
a
2a
Z x
2
2
where erf(x) = √
e−t dt
π 0
Z
x2
2x
2
− 2 + 3
a
a
a
eax
(55)
x3 ex dx = x3 − 3x2 + 6x − 6 ex
(56)
xn eax
n
−
a
a
Z
xn−1 eax dx
(−1)n
Γ[1 + n, −ax],
an+1
Z ∞
where Γ(a, x) =
ta−1 e−t dt
(57)
xn eax dx =
(58)
x
√
√ 2
i π
eax dx = − √ erf ix a
2 a
√
Z
√ 2
π
e−ax dx = √ erf x a
2 a
Z
2
2
1
xe−ax dx = − e−ax
2a
Z
(40)
Z
(41)
x2 eax dx =
xn eax dx =
Z
Z
(49)
xeax dx =
Z
(20)
(21)
(48)
Integrals with Exponentials
Z
(17)
x+a
− 2x (46)
x−a
(31)
1
x
√
dx = sin−1
a
a2 − x2
Z p
b + 2ax p 2
ax2 + bx + cdx =
ax + bx + c
4a
2
p
4ac − b
+
ln 2ax + b + 2 a(ax2 + bx+ c)
8a3/2
x
− 2x (45)
a
x ln(ax + b)dx =
(16)
(23)
p
√
√
dx = x(a + x) − a ln x + x + a
Z
p
x2
1
1 p
√
dx = x x2 ± a2 ∓ a2 ln x + x2 ± a2
2
2
2
2
x ±a
(36)
Z
Z r
1 p 2
1
x
x a − x2 + a2 tan−1 √
2
2
a2 − x2
(30)
(15)
√
x
2
dx = (x ∓ 2a) x ± a
3
x±a
p
x(a − x)
x−a
1p
2ax + b
ln ax2 + bx + c dx =
4ac − b2 tan−1 √
a
4ac − b2
b
+ x ln ax2 + bx + c
(47)
− 2x +
2a
±
Z
(14)
(22)
p
x
dx = − x(a − x) − a tan−1
a−x
Z
(13)
2
(ax + b)5/2
5a
(ax + b)3/2 dx =
√
p
1 p
1
= x x2 ± a2 ± a2 ln x + x2 ± a2
2
2
(29)
a2 dx
Z
Integrals with Roots
√
2
x − adx = (x − a)3/2
3
(44)
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
Z
(11)
Z
b
ln(ax + b)dx = x +
ln(ax + b) − x, a 6= 0
a
Z
(10)
x
x
dx = x − a tan−1
a2 + x2
a
x
1
dx =
ln |ax2 + bx + c|
ax2 + bx + c
2a
b
2ax + b
− √
tan−1 √
a 4ac − b2
4ac − b2
(43)
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
Z
Z
Z
ln ax
1
dx = (ln ax)2
x
2
Z
x2
Z p
2
1
1
a+x
dx =
ln
, a 6= b
(x + a)(x + b)
b−a
b+x
Z
x
a
dx =
+ ln |a + x|
(x + a)2
a+x
Z
(42)
b
b2
x p 3
− 2 +
x3 (ax + b)dx =
x (ax + b)
12a
8a x
3
p
√
b3
(28)
+ 5/2 ln a x + a(ax + b)
8a
(7)
(8)
1
2
2ax + b
dx = √
tan−1 √
2
ax2 + bx + c
4ac − b
4ac − b2
Z
(6)
1
dx = tan−1 x
1 + x2
x
1
dx = ln |a2 + x2 |
a2 + x2
2
Z
Z
(5)
(x + a)n+1
, n 6= −1
n+1
(x + a)n dx =
h
ln axdx = x ln ax − x
(4)
Integrals of Rational Functions
1
1
dx = −
(x + a)2
x+a
Z
Z
(2)
1
1
dx = ln |ax + b|
ax + b
a
Z
√
2
(−2b2 + abx + 3a2 x2 ) ax + b (26)
15a2
2 −ax2
x e
1
dx =
4
r
√
π
x −ax2
erf(x a) −
e
a3
2a
(59)
(60)
(61)
(62)
« 2014. From http://integral-table.com, last revised June 14, 2014. This material is provided as is without warranty or representation about the accuracy, correctness or
suitability of the material for any purpose, and is licensed under the Creative Commons Attribution-Noncommercial-ShareAlike 3.0 United States License. To view a copy of this
license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
Integrals with Trigonometric Functions
Z
Z
Z
Z
Z
1
sin axdx = − cos ax
a
sin2 axdx =
x
sin 2ax
−
2
4a
sec3 x dx =
sec x tan xdx = sec x
Z
2 F1
sin3 axdx = −
3 cos ax
cos 3ax
+
4a
12a
Z
cos axdx =
Z
1
sin ax
a
sin 2ax
x
cos2 axdx = +
2
4a
(85)
1
cosp axdx = −
cos1+p ax×
a(1 + p)
1+p 1 3+p
2
,
,
,
cos
ax
F
2 1
2
2
2
Z
cos3 axdx =
Z
cos ax sin bxdx =
3 sin ax
sin 3ax
+
4a
12a
Z
sec2 x tan xdx =
1
sec2 x
2
1
sec x tan xdx = secn x, n 6= 0
n
n
(86)
(107)
xex sin xdx =
1 x
e (cos x − x cos x + x sin x)
2
(108)
xex cos xdx =
1 x
e (x cos x − sin x + x sin x)
2
(109)
(87)
Z
Z
x
= ln | csc x − cot x| + C
csc xdx = ln tan
2
(66)
(88)
Integrals of Hyperbolic Functions
Z
(67)
1
csc axdx = − cot ax
a
2
(89)
Z
cosh axdx =
Z
(68)
1
1
csc3 xdx = − cot x csc x + ln | csc x − cot x|
2
2
Z
1
csc x cot xdx = − cscn x, n 6= 0
n
Z
sec x csc xdx = ln | tan x|
n
(69)
2
Z
1
x
cos ax + sin ax
a2
a
sin2 x cos xdx =
1
sin3 x
3
cos[(2a − b)x]
cos bx
cos ax sin bxdx =
−
4(2a − b)
2b
cos[(2a + b)x]
−
4(2a + b)
(73)
2x cos ax
a2 x2 − 2
x cos axdx =
+
sin ax
2
a
a3
2
2
Z
Z
cos2 ax sin axdx = −
1
xn cosxdx = − (i)n+1 [Γ(n + 1, −ix)
2
+(−1)n Γ(n + 1, ix)]
Z
(75)
Z
Z
Z
Z
sin 4ax
x
−
8
32a
1
tan axdx = − ln cos ax
a
tan2 axdx = −x +
1
tan ax
a
tann+1 ax
tann axdx =
×
a(1 + n)
n+3
n+1
, 1,
, − tan2 ax
2 F1
2
2
tan3 axdx =
1
1
ln cos ax +
sec2 ax
a
2a
x
sec xdx = ln | sec x + tan x| = 2 tanh−1 tan
2
(96)
(97)
1
(ia)1−n [(−1)n Γ(n + 1, −iax)
2
−Γ(n + 1, ixa)]
(98)
eax sinh bxdx =
 ax
e

 2
[−b cosh bx + a sinh bx]
a − b2
2ax
x

e
−
4a
2
eax tanh bxdx =
 (a+2b)x
h
i

e
a
a


, 1, 2 + , −e2bx
2 F1 1 +


2b
2b
 (a + 2b)
ha
i
1 ax
F
e
,
1,
1E,
−e2bx
−
2
1

a −1 ax 2b

 eax − 2 tan

[e ]


a
Z
1
tanh ax dx = ln cosh ax
a
Z
sec2 axdx =
1
tan ax
a
(112)
a 6= b
(113)
a=b
a 6= b (114)
a=b
(115)
1
[a sin ax cosh bx
a2 + b2
+b cos ax sinh bx]
(116)
cos ax cosh bxdx =
Z
x sin xdx = −x cos x + sin x
Z
(99)
Z
sin ax
x cos ax
+
a
a2
(100)
x2 sin xdx = 2 − x2 cos x + 2x sin x
(101)
x sin axdx = −
(77)
Z
(78)
1
[b cos ax cosh bx+
a2 + b2
a sin ax sinh bx]
(117)
cos ax sinh bxdx =
Z
1
[−a cos ax cosh bx+
a2 + b2
b sin ax sinh bx]
(118)
sin ax cosh bxdx =
Z
(79)
Z
(80)
x2 sin axdx =
2 2
2−a x
2x sin ax
cos ax +
a3
a2
(102)
1
xn sin xdx = − (i)n [Γ(n + 1, −ix) − (−1)n Γ(n + 1, −ix)]
2
(103)
Products of Trigonometric Functions and
Exponentials
(81)
Z
1
[b cosh bx sin ax−
a2 + b2
a cos ax sinh bx]
(119)
sin ax sinh bxdx =
Z
sinh ax cosh axdx =
Z
(82)
ex sin xdx =
1 x
e (sin x − cos x)
2
Z
(83)
ebx sin axdx =
1
ebx (b sin ax − a cos ax)
a2 + b2
1
[−2ax + sinh 2ax]
4a
(120)
(104)
Z
1
[b cosh bx sinh ax
b2 − a2
−a cosh ax sinh bx]
(121)
sinh ax cosh bxdx =
Z
(111)
xn cosaxdx =
sin[2(a − b)x]
x
sin 2ax
−
−
4
8a
16(a − b)
sin[2(a + b)x]
sin 2bx
+
−
(76)
8b
16(a + b)
sin2 ax cos2 axdx =
(95)
(74)
sin2 ax cos2 bxdx =
Z
Z
1
cos3 ax
3a
(94)
Z
Z
Z
x2 cos xdx = 2x cos x + x2 − 2 sin x
(93)
(72)
Z
Z
(92)
Z
x cos axdx =
(110)
eax cosh bxdx =
 ax
e

 2
[a cosh bx − b sinh bx] a 6= b
a − b2
2ax
x

e
+
a=b
4a
2
Z
1
sinh axdx = cosh ax
a
(91)
Z
x cos xdx = cos x + x sin x
1
sinh ax
a
(90)
Products of Trigonometric Functions and
Monomials
(70)
cos[(a − b)x]
cos[(a + b)x]
−
, a 6= b
2(a − b)
2(a + b)
(71)
sin[(2a − b)x]
sin ax cos bxdx = −
4(2a − b)
sin[(2a + b)x]
sin bx
−
+
2b
4(2a + b)
(106)
1
ebx (a sin ax + b cos ax)
a2 + b2
Z
(65)
Z
Z
1 x
e (sin x + cos x)
2
ebx cos axdx =
Z
Z
ex cos xdx =
(64)
sin axdx =
1 1−n 3
,
, , cos2 ax
2
2
2
Z
Z
Z
(84)
(63)
n
1
− cos ax
a
1
1
sec x tan x + ln | sec x + tan x|
2
2
(105)