1. Table of Integrals*
Integrals with Logarithms
Basic Forms
1
x dx =
xn+1
n+1
1
dx = ln |x|
x
(2)
x
dx =
a+x
(1)
n
udv = uv −
vdu
√
x ax + bdx =
x(a + x) − a ln
√
x+
√
x+a
(25)
ln axdx = x ln ax − x
(3)
1
1
dx = ln |ax + b|
ax + b
a
ln ax
1
dx = (ln ax)2
x
2
√
2
(−2b2 + abx + 3a2 x2 ) ax + b (26)
15a2
(42)
(43)
ln(ax + b)dx =
1
x(ax + b)dx = 3/2 (2ax + b) ax(ax + b)
4a
√
(27)
−b2 ln a x + a(ax + b)
(4)
x+
b
a
ln(ax + b) − x, a = 0
ln(x2 + a2 ) dx = x ln(x2 + a2 ) + 2a tan−1
Integrals of Rational Functions
1
1
dx = −
(x + a)2
x+a
(x + a)n dx =
b
b2
x
− 2 +
x3 (ax + b)
12a
8a x
3
√
b3
+ 5/2 ln a x + a(ax + b)
(28)
8a
x3 (ax + b)dx =
(5)
(x + a)n+1
, n = −1
n+1
(6)
n+1
x(x + a)n dx =
(x + a)
((n + 1)x − a)
(n + 1)(n + 2)
1
x2 ± a2 dx = x
2
(7)
1
x2 ± a2 ± a2 ln x +
2
ln(x2 − a2 ) dx = x ln(x2 − a2 ) + a ln
ln ax2 + bx + c dx =
x2 ± a2
(9)
x
1
dx = ln |a2 + x2 |
a2 + x2
2
(10)
a2 − x2 dx =
x
2
a2
x
x
dx = x − a tan−1
+ x2
a
x3
1
1
dx = x2 − a2 ln |a2 + x2 |
2 + x2
a
2
2
1
2
2ax + b
dx = √
tan−1 √
ax2 + bx + c
4ac − b2
4ac − b2
1
1
a+x
dx =
ln
, a=b
(x + a)(x + b)
b−a
b+x
x
a
dx =
+ ln |a + x|
(x + a)2
a+x
ax2
x
1
dx =
ln |ax2 + bx + c|
+ bx + c
2a
2ax + b
b
− √
tan−1 √
a 4ac − b2
4ac − b2
(11)
(13)
x
√
dx =
x2 ± a2
(15)
√
(ax + b)3/2 dx =
√
(16)
(19)
*
«
x
dx = −
a−x
ax2 + bx + cdx =
(33)
eax dx =
(34)
√
a2 − x2
b + 2ax
4a
4ac − b2
ln 2ax + b + 2
8a3/2
xeax dx =
x
1
− 2
a
a
(37)
(24)
eax
(53)
x2
2x
2
− 2 + 3
a
a
a
(54)
eax
(55)
x3 ex dx = x3 − 3x2 + 6x − 6 ex
ax2
x2 eax dx =
(51)
(52)
x2 ex dx = x2 − 2x + 2 ex
(56)
+ bx + c
xn eax dx =
2
ax2 + bx + c
1
1
√
dx = √ ln 2ax + b + 2
a
ax2 + bx + c
xn eax
n
−
a
a
(38)
a(ax2 + bx + c)
dx
x
= √
+ x2 )3/2
a2 a2 + x2
(57)
(58)
ta−1 e−t dt
x
√
√
2
i π
eax dx = − √ erf ix a
2 a
√
√
2
π
e−ax dx = √ erf x a
2 a
2
2
x2 e−ax dx =
1
4
(59)
(60)
1 −ax2
e
2a
(61)
√
π
x −ax2
erf(x a) −
e
a3
2a
(62)
xe−ax dx = −
(40)
(41)
xn−1 eax dx
(−1)n
Γ[1 + n, −ax],
an+1
where Γ(a, x) =
x
1
√
dx =
ax2 + bx + c
a
ax2 + bx + c
b
− 3/2 ln 2ax + b + 2 a(ax2 + bx + c)
2a
(a2
(50)
xex dx = (x − 1)ex
x2 ± a2
(39)
(23)
x(a − x)
x−a
1 ax
e
a
√
√
1 √ ax
i π
xe + 3/2 erf i ax ,
a
2a
x
2
2
where erf(x) = √
e−t dt
π 0
(35)
a(ax2 + bx+ c)
× −3b + 2abx + 8a(c + ax )
√
+3(b3 − 4abc) ln b + 2ax + 2 a
(49)
xeax dx =
ax2 + bx + c
√
1
ax2 + bx + c =
2 a
48a5/2
ln a2 − b2 x2
Integrals with Exponentials
1 2
a ln x +
2
x2 ± a2
(48)
(32)
(21)
√
2a) x ± a
x(a − x) − a tan
x2 ± a2
∞
(22)
−1
(31)
xn eax dx =
(20)
(47)
ln(ax + b)
1
x ln a2 − b2 x2 dx = − x2 +
2
1
a2
x2 − 2
2
b
(36)
2
(ax + b)5/2
5a
2
x
dx = (x
3
x±a
x2
1
dx = x
2
± a2
x2
2
√
ax + b
2b
2x
+
3a
3
3/2
x2 ± a2
x
√
dx = −
a2 − x2
(14)
x
√
1
dx = −2 a − x
a−x
ax + bdx =
1 2
x2 ± a2 dx =
x ± a2
3
(17)
√
2
2
x x − adx = a(x − a)3/2 + (x − a)5/2
3
5
√
1 2
x
a tan−1 √
2
a2 − x2
(30)
a2 − x2 +
1
x
√
dx = sin−1
a
a2 − x2
(18)
2ax + b
4ac − b2 tan−1 √
4ac − b2
bx
1
− x2
2a
4
1
b2
+
x2 − 2
2
a
(12)
+
√
1
√
dx = 2 x ± a
x±a
√
1
x
2
1
√
dx = ln x +
x2 ± a2
Integrals with Roots
√
2
x − adx = (x − a)3/2
3
x+a
− 2x (46)
x−a
x ln(ax + b)dx =
(8)
1
1
x
dx = tan−1
a2 + x2
a
a
x
− 2x (45)
a
b
+ x ln ax2 + bx + c
2a
− 2x +
(29)
1
dx = tan−1 x
1 + x2
1
a
(44)
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USA.

2. Integrals with Trigonometric Functions
sec3 x dx =
1
sin axdx = − cos ax
a
sin2 axdx =
1
1
sec x tan x + ln | sec x + tan x|
2
2
sec x tan xdx = sec x
x
sin 2ax
−
2
4a
sin axdx =
1
sec x tan xdx = secn x, n = 0
n
1 1−n 3
,
, , cos2 ax
2
2
2
(66)
1
sin ax
a
cos ax sin bxdx =
(68)
1
1
csc3 xdx = − cot x csc x + ln | csc x − cot x|
2
2
1
csc x cot xdx = − cscn x, n = 0
n
n
sec x csc xdx = ln | tan x|
(69)
(70)
(107)
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)
(86)
(87)
(88)
(89)
x cos xdx = cos x + x sin x
1
x
cos ax + sin ax
a2
a
x cos axdx =
sin[(2a − b)x]
sin ax cos bxdx = −
4(2a − b)
sin[(2a + b)x]
sin bx
−
+
2b
4(2a + b)
(91)
(92)
(93)
(94)
(95)
(72)
2x cos ax
a2 x2 − 2
x cos axdx =
+
sin ax
2
a
a3
2
1
sin3 x
3
(73)
cos[(2a − b)x]
cos bx
cos ax sin bxdx =
−
4(2a − b)
2b
cos[(2a + b)x]
−
4(2a + b)
1
xn cosxdx = − (i)n+1 [Γ(n + 1, −ix)
2
+(−1)n Γ(n + 1, ix)]
2
1
cos3 ax
3a
eax cosh bxdx =
 ax
 e
 2
[a cosh bx − b sinh bx]
a − b2
2ax
x
e

+
4a
2
sinh axdx =
2
x2 cos xdx = 2x cos x + x2 − 2 sin x
(96)
(97)
eax tanh bxdx =
 (a+2b)x
e
a
a
2bx


 (a + 2b) 2 F1 1 + 2b , 1, 2 + 2b , −e


a
1
, 1, 1E, −e2bx
− eax 2 F1

a −1 ax 2b
 ax

 e − 2 tan [e ]


a
tanh ax dx =
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)
1
(ia)1−n [(−1)n Γ(n + 1, −iax)
2
−Γ(n + 1, ixa)]
(98)
1
tan ax
a
(79)
(80)
1
1
ln cos ax +
sec2 ax
a
2a
sec xdx = ln | sec x + tan x| = 2 tanh−1 tan
1
tan ax
a
a=b
(113)
a=b
a = b (114)
a=b
1
ln cosh ax
a
(115)
1
[a sin ax cosh bx
a2 + b2
+b cos ax sinh bx]
(116)
cos ax cosh bxdx =
(100)
1
[b cos ax cosh bx+
a2 + b2
a sin ax sinh bx]
(117)
(101)
cos ax sinh bxdx =
(81)
x2 sin axdx =
2 2
1
[−a cos ax cosh bx+
a2 + b2
b sin ax sinh bx]
(118)
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)
1
[b cosh bx sin ax−
a2 + b2
a cos ax sinh bx]
(119)
sin ax sinh bxdx =
Products of Trigonometric Functions and
Exponentials
sinh ax cosh axdx =
x
2
(82)
ex sin xdx =
1 x
e (sin x − cos x)
2
(83)
ebx sin axdx =
1
ebx (b sin ax − a cos ax)
a2 + b2
1
[−2ax + sinh 2ax]
4a
(120)
(104)
1
[b cosh bx sinh ax
b2 − a2
−a cosh ax sinh bx]
(121)
sinh ax cosh bxdx =
sec2 axdx =
(112)
sin ax cosh bxdx =
tann+1 ax
×
a(1 + n)
n+3
n+1
, 1,
, − tan2 ax
2
2
tan3 axdx =
sin ax
x cos ax
+
a
a2
x sin axdx = −
(78)
tann axdx =
2 F1
(111)
a=b
(99)
(77)
1
tan axdx = − ln cos ax
a
tan2 axdx = −x +
x sin xdx = −x cos x + sin x
x2 sin xdx = 2 − x2 cos x + 2x sin x
sin 4ax
x
−
8
32a
a=b
xn cosaxdx =
sin2 ax cos2 bxdx =
sin2 ax cos2 axdx =
(110)
1
cosh ax
a
eax sinh bxdx =
 ax
 e
 2
[−b cosh bx + a sinh bx]
a − b2
2ax
x
e

−
4a
2
(74)
(75)
1
sinh ax
a
(90)
Products of Trigonometric Functions and
Monomials
cos[(a − b)x]
cos[(a + b)x]
−
,a = b
2(a − b)
2(a + b)
(71)
cos2 ax sin axdx = −
1
ebx (a sin ax + b cos ax)
a2 + b2
cosh axdx =
3 sin ax
sin 3ax
+
4a
12a
sin2 x cos xdx =
ebx cos axdx =
Integrals of Hyperbolic Functions
1
csc axdx = − cot ax
a
(67)
1
cos1+p ax×
a(1 + p)
1+p 1 3+p
, ,
, cos2 ax
2
2
2
cos3 axdx =
x
= ln | csc x − cot x| + C
csc xdx = ln tan
2
2
sin 2ax
x
cos2 axdx = +
2
4a
2 F1
(106)
xex sin xdx =
(85)
(65)
3 cos ax
cos 3ax
+
4a
12a
cos axdx =
cosp axdx = −
1
sec2 x
2
n
sin3 axdx = −
1 x
e (sin x + cos x)
2
(64)
n
2 F1
ex cos xdx =
(63)
sec2 x tan xdx =
1
− cos ax
a
(84)
(105)