Integral Table 1

Table of Integrals∗ Integrals with Roots

Basic Forms Z

xn dx =

1 xn+1 + c n+1

Z

1 dx = ln x + c x Z Z udv = uv − vdu Z

1 1 dx = ln |ax + b| + c ax + b a

Z

(2)

Z

(3) (4)

Z

Integrals of Rational Functions 1 1 dx = − +c 2 (x + a) x+a

√

(x + a)n+1 + c, n 6= −1 (x + a) dx = n+1 n

(x + a)n+1 ((n + 1)x − a) x(x + a)n dx = +c (n + 1)(n + 2)

Z

x 1 dx = ln |a2 + x2 | + c 2 2 a +x 2

(10)

x2 x dx = x − a tan−1 + c 2 +x a

(11)

x3 1 1 dx = x2 − a2 ln |a2 + x2 | + c + x2 2 2

(12)

a2 a2 Z

Z



√

ax + b + C

(21)

2 (ax + b)5/2 + C 5a

(22)

√ x 2 dx = (x ± 2a) x ± a + C 3 x±a

(23)

(ax + b)3/2 dx =

(8) (9)

Z

2b 2x + 3a 3

(20)

(7)

1 x 1 dx = tan−1 + c a2 + x2 a a

Z

Z

1 dx = tan−1 x + c 1 + x2

(19)



Z r Z

√ 1 dx = −2 a − x + C a−x

ax + bdx =

√

(17) (18)

(6) Z

Z

√

2 (x − a)3/2 + C 3

√ 1 dx = 2 x ± a + C x±a

√

(5) Z

Z

x − adx =

√ 2 2 x x − adx = a(x − a)3/2 + (x − a)5/2 + C 3 5 Z

Z

√

Z (1)

Z r

Z

p x dx = − x(a − x) a−x p x(a − x) +C − a tan−1 x−a

p x dx = x(a + x) a+x √  √ − a ln x + x + a + C

(25)

√ x ax + bdx = √ 2 (−2b2 + abx + 3a2 x2 ) ax + b + C 2 15a

2 2ax + b 1 dx = √ tan−1 √ + C (13) 2 2 ax + bx + c 4ac − b 4ac − b2 Z 1 1 a+x dx = ln , a 6= b (14) (x + a)(x + b) b−a b+x Z x a dx = + ln |a + x| + C (15) 2 (x + a) a+x

(24)

(26)

Z p x(ax + b)dx =

p 1 h (2ax + b) ax(ax + b) 3/2 4a √ i p −b2 ln a x + a(ax + b) + C

Z p

x 1 dx = ln |ax2 + bx + c| 2 ax + bx + c 2a b 2ax + b − √ tan−1 √ + C (16) 2 a 4ac − b 4ac − b2

(27)

 x p 3 b b2 − 2 + x (ax + b) 12a 8a x 3 √ p b3 + 5/2 ln a x + a(ax + b) + C (28) 8a 

x3 (ax + b)dx =

∗ 2007. c From http://integral-table.com, last revised July 5, 2009. This material is provided as is without warranty or representation about the accuracy, correctness or suitability of this material for any purpose. Some restrictions on use and distribution may apply, including the terms of the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. See the web site for details. The formula numbers on this document may be different from the formula numbers on the web page.

1

Integrals with Logarithms Z p 1 p x2 ± a2 dx = x x2 ± a2 2 p 1 2 ± a ln x + x2 ± a2 + C 2 Z p

1 p 2 x a − x2 2 1 x + a2 tan−1 √ +C 2 2 a − x2

Z

Z

Z

x

x2

±

a2 dx


3/2 1 2 x ± a2 +C = 3

ln ax 1 2 dx = (ln ax) + C x 2

(42)

(30)

  b ln(ax + b)dx = x + ln(ax + b) − x + C, a 6= 0 (43) a Z

p

(41)

(29)

a2 − x2 dx =

Z

ln axdx = x ln ax − x + C



ln a2 x2 ± b2 dx = x ln a2 x2 ± b2

(31) +

Z

Z

Z p

p 1 √ dx = ln x + x2 ± a2 + C 2 2 x ±a Z 1 x √ dx = sin−1 + C 2 2 a a −x Z p x √ dx = x2 ± a2 + C x2 ± a2 Z p x √ dx = − a2 − x2 + C a2 − x2

√

x2 1 p dx = x x2 ± a2 2 x2 ± a2 p 1 ∓ a2 ln x + x2 ± a2 + C 2

b + 2ax p 2 ax + bx + c 4a p 4ac − b2 2 + bx+ c) + C + ln + b + 2 a(ax 2ax 8a3/2

Z

p

(32) Z (33)



ln a2 − b2 x2 dx = x ln ar − b2 x2 +

(34) Z

2a bx tan−1 − 2x + C b a

(45)


1p 2ax + b ln ax2 + bx + c dx = 4ac − b2 tan−1 √ a 4ac − b2  
b + x ln ax2 + bx + c + C (46) − 2x + 2a

(35)

Z

bx 1 2 − x 2a 4   1 b2 + x2 − 2 ln(ax + b) + C 2 a

x ln(ax + b)dx =

(36)

Z

(47)

(37)


1 x ln a2 − b2 x2 dx = − x2 + 2  
1 a2 x2 − 2 ln a2 − b2 x2 + C (48) 2 b

ax2 + bx + c =

Integrals with Exponentials Z (38) Z

Z

Z

(44)

ax2 + bx + cdx =

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 + x

x

2b ax tan−1 − 2x + C a b

1 dx = + bx + c p 1 √ ln 2ax + b + 2 a(ax2 + bx + c) + C a √

ax2

x 1p 2 dx = ax + bx + c a + bx + c p b + 3/2 ln 2ax + b + 2 a(ax2 + bx + c) + C 2a

√

(39)

√

(40)

√ √
1 √ ax i π xe dx = xe + 3/2 erf i ax + C, a 2a Z x 2 2 where erf(x) = √ e−t dtet π 0 Z xex dx = (x − 1)ex + C

Z

2

1 ax e +C a

(49)

ax

Z

ax2

eax dx =

xeax dx =



x 1 − a a2



eax + C


x2 ex dx = x2 − 2x + 2 ex + C

(50)

(51) (52) (53)

Z



2 ax

Z

 e

ax

+C

(54)


x3 ex dx = x3 − 3x2 + 6x − 6 ex + C

(55)

x e dx = Z

2 x2 2x − 2 + 3 a a a

(−1)n Γ[1 + n, −ax], an+1 Z ∞ ta−1 e−t dt where Γ(a, x) =

Z

Z

sin2 x cos xdx =

cos[(2a − b)x] cos bx − 4(2a − b) 2b cos[(2a + b)x] − +C 4(2a + b)

Z

cos2 ax sin axdx = −

x

√ √
2 i π eax dx = − √ erf ix a 2 a

Z

Z

(57)

Z

1 sin axdx = − cos ax + C a

sin2 axdx =

x sin 2ax − +C 2 4a

x sin 2ax sin[2(a − b)x] − − 4 8a 16(a − b) sin 2bx sin[2(a + b)x] + − +C 8b 16(a + b)

Z

(58)

sin2 ax cos2 axdx = Z

(59) Z

Z

sin axdx =

Z

 2 F1

sin3 axdx = −

 1 1−n 3 , , , cos2 ax + C 2 2 2

(60)

3 cos ax cos 3ax + +C 4a 12a

(61)

Z

1 sin ax + C a

(62)

x sin 2ax + +C 2 4a

(63)

cos axdx = Z

Z

cos2 axdx =

1 cosp axdx = − cos1+p ax× a(1 + p)   1+p 1 3+p 2 , , , cos ax + C 2 F1 2 2 2 Z

Z

cos3 axdx =

3 sin ax sin 3ax + +C 4a 12a cos[(a − b)x] − 2(a − b) cos[(a + b)x] + C, a 6= b 2(a + b)

Z

Z

tan3 axdx =

(70)

1 tan ax + C a

1 1 ln cos ax + sec2 ax + C a 2a

(71)

(72) (73) (74)

(75)

(76)

Z sec xdx = ln | sec x + tan x| + C  x = 2 tanh−1 tan +C 2 Z

(64)

(65)

Z

sec3 x dx =

sec2 axdx =

1 tan ax + C a

1 1 sec x tan x + ln | sec x + tan x| + C 2 2

(77)

(78)

(79)

Z sec x tan xdx = sec x + C Z

(66) Z

sin[(2a − b)x] sin ax cos bxdx = − 4(2a − b) sin bx sin[(2a + b)x] + − +C 2b 4(2a + b)

1 tan axdx = − ln cos ax + C a

tann+1 ax tann axdx = × a(1 + n)   n+3 n+1 2 , 1, , − tan ax + C 2 F1 2 2

cos ax sin bxdx =

Z

x sin 4ax − +C 8 32a

tan2 axdx = −x +

n

1 − cos ax a

1 cos3 ax + C 3a

(69)

sin2 ax cos2 bxdx =

Integrals with Trigonometric Functions Z

(68)

cos2 ax sin bxdx =

xn eax dx =

(56)

1 sin3 x + C 3

2

Z

(67)

3

1 sec2 x + C 2

(81)

1 secn x + C, n 6= 0 n

(82)

sec2 x tan xdx =

secn x tan xdx =

(80)

x csc xdx = ln tan + C = ln | csc x − cot x| + C 2

(83)

Z

1 csc axdx = − cot ax + C a 2

Products of Trigonometric Functions and Exponentials

(84)

Z Z

1 1 csc3 xdx = − cot x csc x + ln | csc x − cot x| + C (85) 2 2 Z Z

cscn x cot xdx = −

1 cscn x + C, n 6= 0 n

ex sin xdx =

ebx sin axdx = Z

ex cos xdx =

ebx cos axdx =

Products of Trigonometric Functions and Monomials Z Z

Z

x 1 x cos axdx = 2 cos ax + sin ax + C a a

1 ebx (b sin ax − a cos ax) + C (100) a2 + b2 1 x e (sin x + cos x) + C 2

(88) Z (89)

1 ebx (a sin ax + b cos ax) + C (102) a2 + b2

xex sin xdx =

1 x e (cos x − x cos x + x sin x) + C 2

(103)

xex cos xdx =

1 x e (x cos x − sin x + x sin x) + C 2

(104)

Integrals of Hyperbolic Functions

Z

2

2




x cos xdx = 2x cos x + x − 2 sin x + C

(90) Z cosh axdx =

Z

1 sinh ax + C a

(105)

2 2

x2 cos axdx =

Z

Z

2x cos ax a x − 2 + sin ax + C a2 a3

1 xn cosxdx = − (i)n+1 [Γ(n + 1, −ix) 2 +(−1)n Γ(n + 1, ix)] + C

1 (ia)1−n [(−1)n Γ(n + 1, −iax) 2 −Γ(n + 1, ixa)] + C

(91)

Z

(92)

xn cosaxdx =

Z (93)

Z x sin xdx = −x cos x + sin x + C Z x sin axdx = − Z

Z

(101)

(87) Z

x cos xdx = cos x + x sin x + C

(99)

(86)

Z sec x csc xdx = ln | tan x| + C

1 x e (sin x − cos x) + C 2

x cos ax sin ax + +C a a2

(94) Z (95)


x2 sin xdx = 2 − x2 cos x + 2x sin x + C

(96)

2 − a2 x2 2x sin ax cos ax + +C 3 a a2

(97)

x2 sin axdx =

Z

1 xn sin xdx = − (i)n [Γ(n + 1, −ix) 2 −(−1)n Γ(n + 1, −ix)] + C

eax cosh bxdx =  ax e   2 [a cosh bx − b sinh bx] + C a 6= b a − b2 2ax x  e + +C a=b 4a 2 Z 1 sinh axdx = cosh ax + C a eax sinh bxdx =  ax e   2 [−b cosh bx + a sinh bx] + C a − b2 2ax x  e − +C 4a 2

a 6= b (108) a=b

1 [a sin ax cosh bx + b2 +b cos ax sinh bx] + C

cos ax cosh bxdx =

4

(107)

eax tanh bxdx =  (a+2b)x h i a a e  2bx   F 1 + , 1, 2 + , −e 2 1   2b  (a + 2b) h2ba i 1 ax 2bx e F , 1, 1E, −e − + C a 6= b (109) 2 1  a −1 ax 2b  ax     e − 2 tan [e ] + C a=b a Z 1 tanh bxdx = ln cosh ax + C (110) a Z

(98)

(106)

a2

(111)

Z

1 [b cos ax cosh bx+ + b2 a sin ax sinh bx] + C (112)

cos ax sinh bxdx =

Z

Z sinh ax cosh axdx =

a2

Z 1 [−a cos ax cosh bx+ a2 + b2 b sin ax sinh bx] + C (113) 1 [b cosh bx sin ax− a2 + b2 a cos ax sinh bx] + C

sin ax sinh bxdx =

(114)

5

(115)

1 [b cosh bx sinh ax b2 − a2 −a cosh ax sinh bx] + C (116)

sinh ax cosh bxdx =

sin ax cosh bxdx =

Z

1 [−2ax + sinh 2ax] + C 4a