Table of Integrals - Oregon State University



Table of Integrals

Basic Forms

Z

n

1 n+1

x dx =

x

n+1 Z

1

||

dx = ln x

x

Z

Z

udv = uv vdu Z

Zp

2

2

2 2p

x ax + bdx = 2 ( 2b + abx + 3a x ) ax + b (26)

15a

(1)

(2)

Zp

h

p

1

x(ax + b)dx = 3/2 (2ax + b) ax(ax + b)

(3)

4a

2

p

p

i

b ln a x + a(ax + b)

(27)

1

1|

|

dx = ln ax + b

(4)

ax + b a

Integrals of Rational Functions

Z

1

1

2 dx =

(5)

(x + a)

x+a

Z

n+1

n

(x + a)

6

(x + a) dx =

,n = 1

(6)

n+1

Z

n+1

n (x + a) ((n + 1)x a)

x(x + a) dx =

(7)

(n + 1)(n + 2)

Z

1

1

2 dx = tan x

(8)

1+x

Z

1

1

1x

2 2 dx = tan

(9)

a +x

a

a

Z

x

2

2 dx

=

1

|2 ln a

2| +x

(10)

a +x

2

Z

2

x

1x

2 2 dx = x a tan

(11)

a +x

a

Z

3

x

12

2 2 dx = x

1 2 | 2 2| a ln a + x

(12)

a +x

22

Z

1

2

p2

1 p2ax + b

dx =

2 tan

2 (13)

ax + bx + c

4ac b

4ac b

Z

1

1 a+x 6

dx =

ln

, a = b (14)

(x + a)(x + b) b a b + x

Z

Zp 3

2

p

b

b

x

3

x (ax + b)dx =

2+

x (ax + b)

12a 8a x 3

3

pp

b

+ 5/2 ln a x + a(ax + b) (28) 8a

Zp

p

p

2? 2

1

2? 2?1 2

2? 2

x a dx = x x a a ln x + x a

2

2

(29)

Zp 2 a

p

2

1

2

x dx = x a

2

2 12

1p x

x + a tan

2

2

2

ax

(30)

Zp 2? 2

1 2 ? 2 3/2

x x a dx = x a

3

(31)

Z

p

p1

2? 2

2 ? 2 dx = ln x + x a

(32)

xa

Z

p1

1x

2 2 dx = sin

(33)

ax

a

Z

p

px

2? 2

2 ? 2 dx = x a

(34)

xa

Z

p

px 2 2 dx =

2

2

ax

(35)

ax

x 2 dx =

a

|| + ln a + x

(x + a)

a+x

Z

x

2

1 |2

|

dx = ln ax + bx + c

ax + bx + c 2a

(15)

Z

2

p

p

px

1

2? 21 2

2? 2

2

x

? 2 dx a

=

x 2

x

a

a ln x + x 2

a

(36)

pb

1 p2ax + b

2 tan

2 (16)

a 4ac b

4ac b

Integrals with Roots

Zp

p

2

b + 2ax 2

ax + bx + cdx =

ax + bx + c

Zp

2

3/2

x adx = (x a)

4a

2

p

4ac b

2

+

+ 3 2 ln 2ax + b + 2 a(ax + bx c) (37) / 8a

(17)

Zp xx

3

Z

p

p1

?

? dx = 2 x a

xa

Z

p

p1

dx = 2 a x

ax

2

32 /

2

52 /

adx = a(x a) + (x a)

(18) Z p

pp

2

1

2

x ax + bx + c = 5 2 2 a ax + bx + c /

48a

(19)

2

2

3b + 2abx + 8a(c + ax )

3

pp 2

+3(b 4abc) ln b + 2ax + 2 a ax + bx + c (38)

(20)

3

5

Zp

p

2b 2x

ax + bdx = +

ax + b

3a 3 Z

Z

p

(21) p 1

p1

2

2

dx = ln 2ax + b + 2 a(ax + bx + c)

ax + bx + c

a

3/2

2

5/2

(39)

(ax + b) dx = (ax + b)

(22)

5a

Z

p

px

2

?

x

?

dx a

=

(x 3

2a) x

a

Zr

p

x

p 1 x(a

dx = x(a x) a tan

(23)

x) (24)

Z

p

px 2

1

2

dx = ax + bx + c

ax + bx + c a

p

b

2

3/2 ln 2ax + b + 2 a(ax + bx + c)

(40)

2a

ax

xa

Zr

p

x

p p

dx = x(a + x) a ln x + x + a (25)

a+x

Z

dx 2 232=

/

px

22

2

(a + x )

a a +x

(41)

Integrals with Logarithms

Z

ln axdx = x ln ax x

(42)

Z

ln ax 1

2

dx = (ln ax)

(43)

x

2

Z

b ln(ax + b)dx = x +

ln(ax + b)

x, a 6= 0

(44)

a

Z

2

2

2

2

1x

ln(x + a ) dx = x ln(x + a ) + 2a tan

2x (45)

a

Z

2

2

2

2

x+a

ln(x a ) dx = x ln(x a ) + a ln

2x (46)

xa

Z

p

2

1

ln ax + bx + c dx = 4ac

a

2

1 p2ax + b

b tan

2

4ac b

b

2

2x + + x ln ax + bx + c

(47)

2a

Z

bx 1 2

x ln(ax + b)dx =

x

2a 4 2 1 2b

+ x 2 ln(ax + b) (48)

2

a

Z

2

x ln a

22

12

b x dx = x +

2 2

1 2a

2

x 2 ln a

2

b

22

bx

(49)

Integrals with Exponentials

Z

ax

1 ax

e dx = e

a

(50)

Zp

p

p

p

ax

1 ax i

xe dx = xe + 3/2 erf i ax ,

a

2a Z

x

2

p2

t

where erf(x) =

e dt (51)

0

Z

x

x

xe dx = (x 1)e

(52)

Z

ax

x

xe dx =

a Z

1 ax

2e a

(53)

2x

2

x

x e dx = x 2x + 2 e

(54)

Z

2 ax

2 x

x e dx =

a Z

2x 2 ax 2+ 3 e

aa

3x

3

2

x

x e dx = x 3x + 6x 6 e

(55) (56)

Z

n ax

n ax

xe

x e dx =

a

Z

n

n 1 ax

x e dx

a

(57)

Z

n

n ax

( 1)

x e dx = +1 [1 + n, ax],

n

a Z1

(58)

a1 t

where (a, x) = t e dt

x

Z 2

p

p

ax

i p

e dx =

erf ix a

(59)

2a

Z

p

2

p

ax

p

e dx = erf x a

(60)

2a

Z

2

2

ax

1 ax

xe dx = e

(61)

2a

Z

2

2

ax

r p

1

x e dx =

3 erf(x a)

4a

2

x ax e

2a

(62)

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Integrals with Trigonometric Functions

Z

1

sin axdx = cos ax

(63)

a Z

2

x sin 2ax

sin axdx =

(64)

2 4a

Z

n

sin axdx =

1

11 n3 2

cos ax 2F1 ,

, , cos ax

(65)

a

222

Z

3

3 cos ax cos 3ax

sin axdx =

+

(66)

4a

12a

Z

1

cos axdx = sin ax

(67)

a Z

2

x sin 2ax

cos axdx = +

(68)

2 4a

Z

p

cos axdx =

1

1+p

cos ax

a(1 + p)

1+p 1 3+p 2

2 F1

, , , cos ax (69)

222

Z

3

3 sin ax sin 3ax

cos axdx =

+

(70)

4a

12a

Z

3

1

1|

|

sec x dx = sec x tan x + ln sec x + tan x (84)

2

2

Z

sec x tan xdx = sec x

(85)

Z

2

12

sec x tan xdx = sec x

(86)

2 Z

n

1n 6

sec x tan xdx = sec x, n = 0

(87)

n

Z

x

|

|

csc xdx = ln tan = ln csc x cot x + C (88)

2

Z

2

1

csc axdx = cot ax

(89)

a

Z

3

1

1|

|

csc xdx = cot x csc x + ln csc x cot x (90)

2

2

Z

n

1n 6

csc x cot xdx = csc x, n = 0

(91)

n Z

||

sec x csc xdx = ln tan x

(92)

Products of Trigonometric Functions and Monomials

Z

x

1x

e cos xdx = e (sin x + cos x)

2

(106)

Z

bx

1 bx

e cos axdx = 2 2 e (a sin ax + b cos ax) (107)

a +b

Z

x

1x

xe sin xdx = e (cos x x cos x + x sin x)

2

(108)

Z

x

1x

xe cos xdx = e (x cos x sin x + x sin x)

2

(109)

Integrals of Hyperbolic Functions

Z 1

cosh axdx = sinh ax a

Z

ax

e cosh bxdx =

8

><

ax

e

2 2 [a cosh bx

>:

a2ax e

b

x

+

4a 2

Z

6 b sinh bx] a = b

a=b

1 sinh axdx = cosh ax

a

(110)

(111) (112)

Z cos[(a b)x]

cos ax sin bxdx = 2(a b)

cos[(a + b)x] 6 ,a = b

2(a + b)

(71)

Z

2

sin[(2a b)x]

sin ax cos bxdx =

4(2a b)

sin bx sin[(2a + b)x]

+

(72)

2b

4(2a + b)

Z

2

13

sin x cos xdx = sin x

(73)

3

Z

2

cos[(2a b)x] cos bx

cos ax sin bxdx =

4(2a b)

2b

cos[(2a + b)x] (74)

4(2a + b)

Z

2

13

cos ax sin axdx = cos ax

(75)

3a

Z

2

2

x

sin ax cos bxdx =

4

sin 2ax 8a

sin[2(a b)x] 16(a b)

sin 2bx sin[2(a + b)x]

+

(76)

8b

16(a + b)

Z

2

2

x sin 4ax

sin ax cos axdx =

(77)

8 32a Z

1

tan axdx = ln cos ax

(78)

a Z

2

1

tan axdx = x + tan ax

(79)

a

Z

x cos xdx = cos x + x sin x

(93)

Z

1

x

x cos axdx = 2 cos ax + sin ax

(94)

a

a

Z

2

2

x cos xdx = 2x cos x + x 2 sin x (95)

Z

2

22

2x cos ax a x 2

x cos axdx = 2 + 3 sin ax (96)

a

a

Z

n

1

+1 n

x cosxdx = (i) [ (n + 1, ix)

2

n

+( 1) (n + 1, ix)]

(97)

Z

n

1

1 n

n

x cosaxdx = (ia) [( 1) (n + 1, iax)

2

(n + 1, ixa)]

(98)

Z

x sin xdx = x cos x + sin x

Z

x cos ax sin ax

x sin axdx =

+2

a

a

Z

2

2

x sin xdx = 2 x cos x + 2x sin x

(99) (100) (101)

Z

2

22

2 ax

2x sin ax

x sin axdx = 3 cos ax + 2

a

a

(102)

Z

ax

e sinh bxdx =

8

><

ax

e

2 2 [ b cosh bx + a sinh bx]

>:

a2 ax

e

b

x

4a 2

a 6= b a=b

(113)

Z

ax

e tanh bxdx =

8

>>>>><

(a+2b)x

e

(a + 2b)

2

F1

h 1

+

a

, 1, 2 2hb

+

a ,

2b

i

2bx

e i

>>>>>:

ax

e

1

ax

e

2

F1

a , 1, 1E,

2 bx

e

a 1 ax 2b 2 tan [e ]

a Z

1 tanh ax dx = ln cosh ax

a

6 a=b a=b

(114) (115)

Z

1 cos ax cosh bxdx = 2 2 [a sin ax cosh bx

a +b

+b cos ax sinh bx]

(116)

Z

1 cos ax sinh bxdx = 2 2 [b cos ax cosh bx+

a +b

a sin ax sinh bx]

(117)

Z

1 sin ax cosh bxdx = 2 2 [ a cos ax cosh bx+

a +b

b sin ax sinh bx]

(118)

Z

n+1

n

tan ax

tan axdx =

a(1 + n)

n+1 n+3

2

2 F1

, 1, , tan ax

2

2

Z

3

1

12

tan axdx = ln cos ax + sec ax

a

2a

Z

n

1n

n

x sin xdx = (i) [ (n + 1, ix) ( 1) (n + 1, ix)]

2

(103) (80)

Products of Trigonometric Functions and Exponentials

(81)

Z

|

|

1

x

sec xdx = ln sec x + tan x = 2 tanh tan (82)

2

Z

x

1x

e sin xdx = e (sin x cos x)

2

(104)

Z

2

1

sec axdx = tan ax

a

Z

bx

1 bx

(83)

e sin axdx = 2 2 e (b sin ax a cos ax) (105)

a +b

Z 1

sin ax sinh bxdx = 2 2 [b cosh bx sin ax a +b

a cos ax sinh bx]

(119)

Z 1

sinh ax cosh axdx = [ 2ax + sinh 2ax] 4a

(120)

Z

1 sinh ax cosh bxdx = 2 2 [b cosh bx sinh ax

ba

a cosh ax sinh bx]

(121)

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