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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
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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)
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