INtegral

1. Table of Integrals∗
Basic Forms Integrals with Roots
1 √ 2
xn dx = xn+1 + c (1) x − adx = (x − a)3/2 + C (17)
n+1 3
1 1 √
dx = ln x + c (2) √ dx = 2 x ± a + C (18)
x x±a
1 √
udv = uv − vdu (3) √ dx = 2 a − x + C (19)
a−x
1 1
dx = ln |ax + b| + c (4)
ax + b a √ 2 2
x x − adx = a(x − a)3/2 + (x − a)5/2 + C (20)
3 5
Integrals of Rational Functions
√ 2b 2x √
1 1 ax + bdx = + ax + b + C (21)
dx = − +c (5) 3a 3
(x + a)2 x+a
2
(x + a)n+1 (ax + b)3/2 dx = (ax + b)5/2 + C (22)
n
(x + a) dx = + c, n = −1 (6) 5a
n+1
x 2 √
√ dx = (x ± 2a) x ± a + C (23)
x±a 3
(x + a)n+1 ((n + 1)x − a)
x(x + a)n dx = +c (7)
(n + 1)(n + 2)
x
1 dx = − x(a − x)
dx = tan−1 x + c (8) a−x
1 + x2
x(a − x)
− a tan−1 +C (24)
1 1 x x−a
dx = tan−1 + c (9)
a2 + x2 a a
x 1 x
dx = ln |a2 + x2 | + c (10) dx = x(a + x)
a2 +x 2 2 a+x
√ √
x2 x − a ln x + x + a + C (25)
dx = x − a tan−1 + c (11)
a2 +x 2 a
x3 1 1 √
dx = x2 − a2 ln |a2 + x2 | + c (12)
a2 + x2 2 2 x ax + bdx =
2 √
1 2 2ax + b 2
(−2b2 + abx + 3a2 x2 ) ax + b + C (26)
dx = √ tan−1 √ + C (13) 15a
ax 2 + bx + c 2
4ac − b 4ac − b2
1 1 a+x 1
dx = ln , a=b (14) x(ax + b)dx = (2ax + b) ax(ax + b)
(x + a)(x + b) b−a b+x 4a3/2
√
x a −b2 ln a x + a(ax + b) + C (27)
2
dx = + ln |a + x| + C (15)
(x + a) a+x
x 1 b b2 x
dx = ln |ax2 + bx + c| x3 (ax + b)dx = − 2 + x3 (ax + b)
ax2 + bx + c 2a 12a 8a x 3
b 2ax + b b3 √
− √ tan−1 √ + C (16) + 5/2 ln a x + a(ax + b) + C (28)
a 4ac − b 2 4ac − b2 8a
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1

2. Integrals with Logarithms
1
x2 ± a2 dx = x x2 ± a2
2 ln axdx = x ln ax − x + C (41)
1
± a2 ln x + x2 ± a2 + C (29)
2 ln ax 1 2
dx = (ln ax) + C (42)
x 2
1
a2 − x2 dx = x a2 − x2 b
2 ln(ax + b)dx = x+ ln(ax + b) − x + C, a = 0 (43)
1 x a
+ a2 tan−1 √ +C (30)
2 a2 − x2
1 2 3/2 ln a2 x2 ± b2 dx = x ln a2 x2 ± b2
x x2 ± a2 dx = x ± a2 +C (31)
3 2b ax
+ tan−1 − 2x + C (44)
1 a b
√ dx = ln x + x2 ± a2 + C (32)
x 2 ± a2
1 x ln a2 − b2 x2 dx = x ln ar − b2 x2
√ dx = sin−1 + C (33)
a2 − x2 a
2a bx
+ tan−1 − 2x + C (45)
x b a
√ dx = x2 ± a2 + C (34)
x2 ± a2
x 1 2ax + b
√ dx = − a2 − x2 + C (35) ln ax2 + bx + c dx = 4ac − b2 tan−1 √
a2 − x2 a 4ac − b2
b
− 2x + + x ln ax2 + bx + c + C (46)
2a
x2 1
√ dx = x x2 ± a2
x 2 ± a2 2
1 2 bx 1 2
a ln x + x2 ± a2 + C (36) x ln(ax + b)dx = − x
2 2a 4
1 b2
+ x2 − 2 ln(ax + b) + C (47)
2 a
b + 2ax
ax2 + bx + cdx = ax2 + bx + c
4a
1
4ac − b2 x ln a2 − b2 x2 dx = − x2 +
+ ln 2ax + b + 2 a(ax2 + bx+ c) + C (37) 2
8a3/2
1 a2
x2 − 2 ln a2 − b2 x2 + C (48)
2 b
1 √
x ax2 + bx + c = 5/2
2 a ax2 + bx + c Integrals with Exponentials
48a
− 3b2 + 2abx + 8a(c + ax2 )
√ 1 ax
eax dx = e +C (49)
+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + x (38) a
√
√ ax 1 √ ax i π √
1 xe dx = xe + 3/2 erf i ax + C,
√ dx = a 2a
x
ax2 + bx + c 2 2
1 where erf(x) = √ e−t dtet (50)
√ ln 2ax + b + 2 a(ax2 + bx + c) + C (39) π 0
a
xex dx = (x − 1)ex + C (51)
x 1
√ dx = ax2 + bx + c x 1
ax2 + bx + c a xeax dx = − eax + C (52)
b a a2
+ 3/2 ln 2ax + b + 2 a(ax2 + bx + c) + C (40)
2a
x2 ex dx = x2 − 2x + 2 ex + C (53)
2

3. x2 2x 2 1
x2 eax dx = − 2 + 3 eax + C (54) sin2 x cos xdx = sin3 x + C (68)
a a a 3
x3 ex dx = x3 − 3x2 + 6x − 6 ex + C (55) cos[(2a − b)x] cos bx
cos2 ax sin bxdx = −
4(2a − b) 2b
cos[(2a + b)x]
1 − +C (69)
xn eax dx = (−1)n Γ[1 + n, −ax], 4(2a + b)
a
∞
where Γ(a, x) = ta−1 e−t dt (56) 1
cos2 ax sin axdx = − cos3 ax + C (70)
x 3a
√
2 i π √
eax dx = − √ erf ix a (57) x sin 2ax sin[2(a − b)x]
2 a sin2 ax cos2 bxdx = − −
4 8a 16(a − b)
Integrals with Trigonometric Functions sin 2bx sin[2(a + b)x]
+ − +C (71)
8b 16(a + b)
1
sin axdx = − cos ax + C (58) x sin 4ax
a sin2 ax cos2 axdx = − +C (72)
8 32a
x sin 2ax
sin2 axdx = − +C (59) 1
2 4a tan axdx = − ln cos ax + C (73)
a
1
n tan2 axdx = −x + tan ax + C (74)
sin axdx = a
1 1 1−n 3
− cos ax 2 F1 , , , cos2 ax + C (60) tann+1 ax
a 2 2 2 tann axdx = ×
a(1 + n)
3 cos ax cos 3ax n+1 n+3
sin3 axdx = − + +C (61) 2 F1 , 1, , − tan2 ax + C (75)
4a 12a 2 2
1 1 1
cos axdx = sin ax + C (62) tan3 axdx = ln cos ax + sec2 ax + C (76)
a a 2a
x sin 2ax
cos2 axdx = + +C (63)
2 4a
sec xdx = ln | sec x + tan x| + C
x
1 = 2 tanh−1 tan +C (77)
cosp axdx = − cos1+p ax× 2
a(1 + p)
1+p 1 3+p 1
2 F1 , , , cos2 ax + C (64) sec2 axdx = tan ax + C (78)
2 2 2 a
3 sin ax sin 3ax
cos3 axdx = + +C (65) 1 1
4a 12a sec3 xdx = sec x tan x + ln | sec x tan x| + C (79)
2 2
cos[(a − b)x]
cos ax sin bxdx = − sec x tan xdx = sec x + C (80)
2(a − b)
cos[(a + b)x]
+ C, a = b (66) 1
2(a + b) sec2 x tan xdx = sec2 x + C (81)
2
1
2 sin[(2a − b)x] secn x tan xdx = secn x + C, n = 0 (82)
sin ax cos bxdx = − n
4(2a − b)
sin bx sin[(2a + b)x]
+ − +C (67) x
2b 4(2a + b) csc xdx = ln tan + C = ln | csc x − cot x| + C (83)
2
3

4. Products of Trigonometric Functions and
2 1 Exponentials
csc axdx = − cot ax + C (84)
a
1 x
ex sin xdx = e (sin x − cos x) + C (99)
1 1 2
csc3 xdx = − cot x csc x + ln | csc x − cot x| + C (85)
2 2
1
ebx sin axdx = ebx (b sin ax − a cos bx) + C (100)
1 a2 + b2
cscn x cot xdx = − cscn x + C, n = 0 (86)
n 1 x
ex cos xdx = e (sin x + cos x) + C (101)
2
sec x csc xdx = ln | tan x| + C (87)
1
ebx cos axdx = ebx (a sin ax + b cos ax) + C (102)
Products of Trigonometric Functions and Monomials a2 + b2
1 x
xex sin xdx = e (cos x − x cos x + x sin x) + C (103)
x cos xdx = cos x + x sin x + C (88) 2
1 x 1 x
x cos axdx = cos ax + sin ax + C (89) xex cos xdx = e (x cos x − sin x + x sin x) + C (104)
a2 a 2
Integrals of Hyperbolic Functions
2 2
x cos xdx = 2x cos x + x − 2 sin x + C (90)
1
cosh axdx = sinh ax + C (105)
a
2 2
2x cos ax a x − 2
x2 cos axdx = + sin ax + C (91)
a2 a3
eax cosh bxdx =
 ax
 e [a cosh bx − b sinh bx] + C a=b
1  2
a − b2
xn cosxdx = − (i)n+1 [Γ(n + 1, −ix) 2ax (106)
2 e x
 + +C a=b
+(−1)n Γ(n + 1, ix)] + C (92) 4a 2
1
sinh axdx = cosh ax + C (107)
a
1
xn cosaxdx = (ia)1−n [(−1)n Γ(n + 1, −iax)
2 eax sinh bxdx =
−Γ(n + 1, ixa)] + C (93)  ax
 e
 2 [−b cosh bx + a sinh bx] + C a=b
a − b2 (108)
2ax
e x
x sin xdx = −x cos x + sin x + C (94)  − +C a=b
4a 2
x cos ax sin ax
x sin axdx = − + +C (95) eax tanh bxdx =
a a2
 (a+2b)x
e a a 2bx
 (a + 2b) 2 F1 1 + 2b , 1, 2 + 2b , −e

x2 sin xdx = 2 − x2 cos x + 2x sin x + C

(96) 

1 a
− eax 2 F1 , 1, 1E, −e2bx + C a = b (109)
 a −1 ax 2b
 eax − 2 tan [e ]


2 − a2 x2 +C a=b

2x sin ax 
x2 sin axdx = 3
cos ax + +C (97) a
a a3
1
tanh bxdx = ln cosh ax + C (110)
a
1
xn sin xdx = − (i)n [Γ(n + 1, −ix) 1
2 cos ax cosh bxdx = [a sin ax cosh bx
−(−1)n Γ(n + 1, −ix)] + C (98) a2 + b2
+b cos ax sinh bx] + C (111)
4

5. 1 1
cos ax sinh bxdx = [b cos ax cosh bx+ sinh ax cosh axdx = [−2ax + sinh 2ax] + C (115)
a2+ b2 4a
a sin ax sinh bx] + C (112)
1
1 sinh ax cosh bxdx = [b cosh bx sinh ax
sin ax cosh bxdx = [−a cos ax cosh bx+ b2 − a2
a2 + b2
−a cosh ax sinh bx] + C (116)
b sin ax sinh bx] + C (113)
1
sin ax sinh bxdx = [b cosh bx sin ax−
a2 + b2
a cos ax sinh bx] + C (114)
5