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Table of Integrals
Basic Forms
x n dx =
n + 1
x n+
x
dx = ln x + c (2)
udv = uv −
vdu (3)
ax + b
dx =
a
ln |ax + b| + c (4)
Integrals of Rational Functions
(x + a)^2
dx = −
x + a
(x + a) n dx =
(x + a) n+
n + 1
x(x + a) n dx =
(x + a) n+ ((n + 1)x − a)
(n + 1)(n + 2)
1 + x^2
dx = tan − 1 x + c (8)
a^2 + x^2
dx =
a
tan − 1 x a
x
a^2 + x^2
dx =
ln |a 2
x 2
a^2 + x^2
dx = x − a tan
− 1 x a
x^3
a^2 + x^2
dx =
x 2 −
a 2 ln |a 2
ax^2 + bx + c
dx =
4 ac − b^2
tan − 1 2 ax^ +^ b √ 4 ac − b^2
+ C (13)
(x + a)(x + b)
dx =
b − a
ln
a + x
b + x
, a 6 = b (14)
x
(x + a)^2
dx =
a
a + x
x
ax^2 + bx + c
dx =
2 a
ln |ax
2
b
a
4 ac − b^2
tan − (^1) √^2 ax^ +^ b 4 ac − b^2
+ C (16)
Integrals with Roots
x − adx =
(x − a) 3 / 2
x ± a
dx = 2
x ± a + C (18)
a − x
dx = − 2
a − x + C (19)
x
x − adx =
a(x − a) 3 / 2
(x − a) 5 / 2
ax + bdx =
2 b
3 a
2 x
3
ax + b + C (21)
(ax + b) 3 / 2 dx =
5 a
(ax + b) 5 / 2
x √ x ± a
dx =
(x ∓ 2 a)
x ± a + C (23)
∫ √^
x
a − x
dx = −
x(a − x)
− a tan − 1
x(a − x)
x − a
+ C (24)
∫ √^
x
a + x
dx =
x(a + x)
− a ln
[√
x +
x + a
]
+ C (25)
x
ax + bdx =
15 a^2
(− 2 b 2
ax + b + C (26)
x(ax + b)dx =
4 a^3 /^2
[
(2ax + b)
ax(ax + b)
−b 2 ln
∣a
x +
a(ax + b)
]
+ C (27)
x^3 (ax + b)dx =
[
b
12 a
b 2
8 a^2 x
x
3
]
x^3 (ax + b)
b^3
8 a^5 /^2
ln
∣a
x +
a(ax + b)
∣ +^ C^ (28)
∗« 2009. From http://integral-table.com, last revised February 17, 2010. This material is provided as is without warranty or representation
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x^2 ± a^2 dx =
x
x^2 ± a^2
a 2 ln
∣x^ +^
x^2 ± a^2
∣ +^ C^ (29)
a^2 − x^2 dx =
x
a^2 − x^2
a 2 tan − 1 x √ a^2 − x^2
+ C (30)
x
x^2 ± a^2 dx =
x
2 ± a
2 )^3 /^2
+ C (31)
x^2 ± a^2
dx = ln
∣x^ +^
x^2 ± a^2
∣ +^ C^ (32)
a^2 − x^2
dx = sin − 1 x a
+ C (33)
x √ x^2 ± a^2
dx =
x^2 ± a^2 + C (34)
x √ a^2 − x^2
dx = −
a^2 − x^2 + C (35)
x^2 √ x^2 ± a^2
dx =
x
x^2 ± a^2
a 2 ln
∣x^ +^
x^2 ± a^2
∣ +^ C^ (36)
ax^2 + bx + cdx =
b + 2ax
4 a
ax^2 + bx + c
4 ac − b^2
8 a^3 /^2
ln
∣ 2 ax + b + 2
a(ax^2 + bx+c)
∣ + C (37)
x
ax^2 + bx + c =
48 a^5 /^2
a
ax^2 + bx + c
3 b 2
+3(b 3 − 4 abc) ln
∣b^ + 2ax^ + 2
a
ax^2 + bx + x
ax^2 + bx + c
dx =
a
ln
∣ 2 ax + b + 2
a(ax^2 + bx + c)
∣ + C (39)
x √ ax^2 + bx + c
dx =
a
ax^2 + bx + c
b
2 a^3 /^2
ln
∣ 2 ax + b + 2
a(ax^2 + bx + c)
∣ + C (40)
Integrals with Logarithms
ln axdx = x ln ax − x + C (41)
ln ax
x
dx =
(ln ax)
2
ln(ax + b)dx =
x +
b
a
ln(ax + b) − x + C, a 6 = 0 (43)
ln
a 2 x 2 ± b 2
dx = x ln
a 2 x 2 ± b 2
2 b
a
tan − 1 ax b
− 2 x + C (44)
ln
a 2 − b 2 x 2
dx = x ln
ar − b 2 x 2
2 a
b
tan − 1 bx a
− 2 x + C (45)
ln
ax 2
dx =
a
4 ac − b^2 tan − (^1) √^2 ax^ +^ b 4 ac − b^2
− 2 x +
b
2 a
ln
ax 2
+ C (46)
x ln(ax + b)dx =
bx
2 a
x 2
x
2 −
b 2
a^2
ln(ax + b) + C (47)
x ln
a 2 − b 2 x 2
dx = −
x 2
x 2 −
a 2
b^2
ln
a 2 − b 2 x 2
+ C (48)
Integrals with Exponentials
e ax dx =
a
e ax
xe
ax dx =
a
xe
ax
i
π
2 a^3 /^2
erf
i
ax
+ C,
where erf(x) =
π
∫ (^) x
0
e −t^2 dtet (50)
xe x dx = (x − 1)e x
xe ax dx =
x
a
a^2
e ax
x 2 e x dx =
x 2 − 2 x + 2
e x
csc xdx = ln
∣tan
x
2
∣ + C = ln | csc x − cot x| + C (85)
csc 2 axdx = −
a
cot ax + C (86)
csc 3 xdx = −
cot x csc x +
ln | csc x − cot x| + C (87)
csc n x cot xdx = −
n
csc n x + C, n 6 = 0 (88)
sec x csc xdx = ln | tan x| + C (89)
Products of Trigonometric Functions and Monomials
x cos xdx = cos x + x sin x + C (90)
x cos axdx =
a^2
cos ax +
x
a
sin ax + C (91)
x 2 cos xdx = 2x cos x +
x 2 − 2
sin x + C (92)
x 2 cos axdx =
2 x cos ax
a^2
a 2 x 2 − 2
a^3
sin ax + C (93)
x n cosxdx = −
(i) n+ [Γ(n + 1, −ix)
n Γ(n + 1, ix)] + C (94)
x n cosaxdx =
(ia) 1 −n [(−1) n Γ(n + 1, −iax)
−Γ(n + 1, ixa)] + C (95)
x sin xdx = −x cos x + sin x + C (96)
x sin axdx = −
x cos ax
a
sin ax
a^2
+ C (97)
x 2 sin xdx =
2 − x 2
cos x + 2x sin x + C (98)
x 2 sin axdx =
2 − a^2 x^2
a^3
cos ax +
2 x sin ax
a^2
+ C (99)
x n sin xdx = −
(i) n [Γ(n + 1, −ix)
n Γ(n + 1, −ix)] + C (100)
Products of Trigonometric Functions and Exponentials
e
x sin xdx =
e
x (sin x − cos x) + C (101)
e bx sin axdx =
a^2 + b^2
e bx (b sin ax − a cos ax) + C (102)
e x cos xdx =
e x (sin x + cos x) + C (103)
e bx cos axdx =
a^2 + b^2
e bx (a sin ax + b cos ax) + C (104)
xe x sin xdx =
e x (cos x − x cos x + x sin x) + C (105)
xe x cos xdx =
e x (x cos x − sin x + x sin x) + C (106)
Integrals of Hyperbolic Functions
cosh axdx =
a
sinh ax + C (107)
e ax cosh bxdx =
eax
a^2 − b^2
[a cosh bx − b sinh bx] + C a 6 = b
e 2 ax
4 a
x
2
sinh axdx =
a
cosh ax + C (109)
e ax sinh bxdx =
e ax
a^2 − b^2
[−b cosh bx + a sinh bx] + C a 6 = b
e^2 ax
4 a
x
2
e
ax tanh bxdx =
e (a+2b)x
(a + 2b)
2 F 1
[
a
2 b
a
2 b
, −e 2 bx
]
a
e ax 2 F 1
[
a
2 b
, 1 , 1 E, −e 2 bx
]
eax^ − 2 tan−^1 [eax]
a
tanh bxdx =
a
ln cosh ax + C (112)
cos ax cosh bxdx =
a^2 + b^2
[a sin ax cosh bx
+b cos ax sinh bx] + C (113)
cos ax sinh bxdx =
a^2 + b^2
[b cos ax cosh bx+
a sin ax sinh bx] + C (114)
sin ax cosh bxdx =
a^2 + b^2
[−a cos ax cosh bx+
b sin ax sinh bx] + C (115)
sin ax sinh bxdx =
a^2 + b^2
[b cosh bx sin ax−
a cos ax sinh bx] + C (116)
sinh ax cosh axdx =
4 a
[− 2 ax + sinh 2ax] + C (117)
sinh ax cosh bxdx =
b^2 − a^2
[b cosh bx sinh ax
−a cosh ax sinh bx] + C (118)
