Table of Integrals
Table of Integrals
Basic Forms
xndx = 1 xn+1
(1)
n+1
1 dx = ln |x|
(2)
x
udv = uv - vdu
(3)
1 dx = 1 ln |ax + b|
(4)
ax + b
a
Integrals of Rational Functions
1
1
(x + a)2 dx = - x + a
(5)
(x + a)ndx
=
(x + a)n+1 ,n
=
-1
(6)
n+1
x(x + a)ndx = (x + a)n+1((n + 1)x - a)
(7)
(n + 1)(n + 2)
x ax + bdx =
2 15a2
(-2b2
+
abx
+
3a2x2
) ax
+
b
(26)
1 x(ax + b)dx = 4a3/2 (2ax + b) ax(ax + b)
-b2 ln
a x+
a(ax + b)
(27)
x3(ax + b)dx =
b
b2
x
12a - 8a2x + 3
x3(ax + b)
b3
+ 8a5/2 ln a x + a(ax + b) (28)
x2 ? a2dx = 1 x x2 ? a2 ? 1 a2 ln x +
2
2
x2 ? a2 (29)
Integrals with Logarithms
ln axdx = x ln ax - x
(42)
ln ax dx = 1 (ln ax)2
(43)
x
2
ln(ax + b)dx = x + b ln(ax + b) - x, a = 0 (44) a
ln(x2 + a2) dx = x ln(x2 + a2) + 2a tan-1 x - 2x (45) a
ln(x2 - a2) dx = x ln(x2 - a2) + a ln x + a - 2x (46) x-a
ln
ax2 + bx + c
1 dx =
4ac - b2 tan-1 2ax + b
a
4ac - b2
- 2x + b + x ln ax2 + bx + c
(47)
2a
1
1 + x2
dx
=
tan-1
x
(8)
a2
1 + x2 dx
=
1 a
tan-1
x a
(9)
a2
x + x2 dx
=
1 2
ln |a2
+ x2|
(10)
x2 dx = x - a tan-1 x
(11)
a2 + x2
a
x3 a2 + x2 dx
=
1 x2 2
-
1 a2 2
ln |a2
+
x2|
(12)
ax2
1
dx
+ bx + c
=
2
4ac - b2
tan-1
2ax + b
4ac - b2
(13)
1
1 a+x
dx =
ln
, a = b (14)
(x + a)(x + b)
b-a b+x
x
a
dx =
+ ln |a + x|
(15)
(x + a)2
a+x
ax2
x dx
+ bx + c
=
1 2a
ln |ax2
+ bx + c|
-b
tan-1 2ax + b (16)
a 4ac - b2
4ac - b2
Integrals with Roots
x - adx
=
2 (x - a)3/2
(17)
3
1
dx = 2 x ? a
(18)
x?a
1
dx = -2 a - x
(19)
a-x
x x-
adx
=
2 a(x
-
a)3/2
+
2 (x
- a)5/2
(20)
3
5
2b 2x
ax + bdx = +
ax + b
(21)
3a 3
(ax + b)3/2dx = 2 (ax + b)5/2
(22)
5a
x
dx
=
2 (x
2a) x
?
a
(23)
x?a
3
x dx = - x(a - x) - a tan-1
x(a - x) (24)
a-x
x-a
a2 - x2dx = 1 x a2 - x2 + 1 a2 tan-1 x
2
2
a2 - x2
(30)
x x2 ? a2dx = 1 x2 ? a2 3/2 3
(31)
1
dx = ln x + x2 ? a2
(32)
x2 ? a2
1 dx = sin-1 x
(33)
a2 - x2
a
x dx = x2 ? a2
(34)
x2 ? a2
x dx = - a2 - x2
(35)
a2 - x2
x2
1 dx = x
x2 ? a2 1 a2 ln x +
x2 ? a2
2
2
x2 ? a2 (36)
ax2 + bx + cdx = b + 2ax ax2 + bx + c 4a
4ac - b2
+
ln 2ax + b + 2
a(ax2 + bx+c)
(37)
8a3/2
x ax2 + bx + c = 1
2 a ax2 + bx + c
48a5/2
? -3b2 + 2abx + 8a(c + ax2)
+3(b3 - 4abc) ln
b + 2ax + 2 a
ax2 + bx + c
(38)
1
dx = 1 ln 2ax + b + 2 a(ax2 + bx + c)
ax2 + bx + c
a
(39)
x
1 dx =
ax2 + bx + c
ax2 + bx + c
a
b - 2a3/2 ln 2ax + b + 2
a(ax2 + bx + c)
(40)
x ln(ax + b)dx = bx - 1 x2 2a 4
+ 1 x2 - b2 ln(ax + b)
(48)
2
a2
x ln a2 - b2x2 dx = - 1 x2+ 2
1 x2 - a2
2
b2
ln a2 - b2x2
(49)
Integrals with Exponentials
eaxdx = 1 eax a
(50)
xeaxdx
=
1 xeax a
+
i 2a3/2 erf
i ax
,
where erf(x) = 2 x e-t2 dt
(51)
0
xexdx = (x - 1)ex
(52)
xeaxdx =
x1 a - a2
eax
(53)
x2exdx = x2 - 2x + 2 ex
(54)
x2eaxdx =
x2 2x 2 -+
eax
a a2 a3
x3exdx = x3 - 3x2 + 6x - 6 ex
xneax dx = xneax - n aa
xn-1eax dx
(55) (56) (57)
xneax
dx
=
(-1)n an+1 [1
+
n, -ax],
where (a, x) =
ta-1e-t dt
x
eax2 dx = - i erf
ix a
2a
e-ax2 dx =
erf
xa
2a
xe-ax2 dx = - 1 e-ax2 2a
(58)
(59) (60) (61)
x
dx = x(a + x) - a ln x + x + a (25)
a+x
dx
x
=
(a2 + x2)3/2 a2 a2 + x2
(41)
x2e-ax2 dx = 1
erf(x a)
-
x e-ax2
4 a3
2a
(62)
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Integrals with Trigonometric Functions
sin axdx = - 1 cos ax
(63)
a
sin2 axdx = x - sin 2ax
(64)
2 4a
sinn axdx =
1
- cos ax a
2 F1
1 , 1 - n , 3 , cos2 ax 222
(65)
sin3
axdx
=
3 cos ax -
+
cos 3ax
(66)
4a
12a
cos axdx = 1 sin ax
(67)
a
cos2 axdx = x + sin 2ax
(68)
2 4a
cosp axdx = - 1 cos1+p ax? a(1 + p)
2 F1
1 + p , 1 , 3 + p , cos2 ax 222
(69)
cos3 axdx = 3 sin ax + sin 3ax
(70)
4a
12a
cos ax sin bxdx = cos[(a - b)x] - cos[(a + b)x] , a = b
2(a - b)
2(a + b)
(71)
sin2 ax cos bxdx = - sin[(2a - b)x] 4(2a - b)
+ sin bx - sin[(2a + b)x]
(72)
2b
4(2a + b)
sin2 x cos xdx = 1 sin3 x
(73)
3
cos2 ax sin bxdx = cos[(2a - b)x] - cos bx
4(2a - b)
2b
cos[(2a + b)x]
-
(74)
4(2a + b)
cos2 ax sin axdx = - 1 cos3 ax
(75)
3a
sin2 ax cos2 bxdx = x - sin 2ax - sin[2(a - b)x]
4 8a
16(a - b)
+ sin 2bx - sin[2(a + b)x]
(76)
8b
16(a + b)
sin2 ax cos2 axdx = x - sin 4ax
(77)
8 32a
tan axdx = - 1 ln cos ax
(78)
a
tan2 axdx = -x + 1 tan ax
(79)
a
tann axdx
=
tann+1 ax ?
a(1 + n)
2 F1
n + 1 , 1, n + 3 , - tan2 ax
2
2
(80)
tan3 axdx = 1 ln cos ax + 1 sec2 ax
(81)
a
2a
sec xdx = ln | sec x + tan x| = 2 tanh-1 tan x (82) 2
sec2 axdx = 1 tan ax
(83)
a
sec3 x dx = 1 sec x tan x + 1 ln | sec x + tan x| (84)
2
2
sec x tan xdx = sec x
(85)
sec2 x tan xdx = 1 sec2 x
(86)
2
secn x tan xdx = 1 secn x, n = 0
(87)
n
x csc xdx = ln tan = ln | csc x - cot x| + C (88)
2
csc2 axdx = - 1 cot ax
(89)
a
csc3 xdx = - 1 cot x csc x + 1 ln | csc x - cot x| (90)
2
2
cscn x cot xdx = - 1 cscn x, n = 0
(91)
n
sec x csc xdx = ln | tan x|
(92)
Products of Trigonometric Functions and Monomials
x cos xdx = cos x + x sin x
(93)
1
x
x cos axdx = a2 cos ax + a sin ax
(94)
x2 cos xdx = 2x cos x + x2 - 2 sin x
(95)
x2 cos axdx
=
2x cos ax a2
+
a2x2 - 2 a3
sin ax
(96)
xncosxdx = - 1 (i)n+1 [(n + 1, -ix) 2
+(-1)n(n + 1, ix)]
(97)
xncosaxdx = 1 (ia)1-n [(-1)n(n + 1, -iax) 2
-(n + 1, ixa)]
(98)
x sin xdx = -x cos x + sin x
x cos ax sin ax
x sin axdx = - a
+ a2
x2 sin xdx = 2 - x2 cos x + 2x sin x
(99) (100) (101)
x2 sin axdx = 2 - a2x2 cos ax + 2x sin ax
a3
a2
(102)
xn sin xdx = - 1 (i)n [(n + 1, -ix) - (-1)n(n + 1, -ix)] 2 (103)
Products of Trigonometric Functions and Exponentials
ex sin xdx = 1 ex(sin x - cos x) 2
(104)
ebx
sin axdx
=
a2
1 +
b2 ebx(b sin ax
-
a cos ax)
(105)
ex cos xdx = 1 ex(sin x + cos x) 2
(106)
ebx cos axdx = 1 ebx(a sin ax + b cos ax) (107) a2 + b2
xex sin xdx = 1 ex(cos x - x cos x + x sin x) (108) 2
xex cos xdx = 1 ex(x cos x - sin x + x sin x) (109) 2
Integrals of Hyperbolic Functions
cosh axdx = 1 sinh ax a
(110)
eax cosh bxdx =
eax
a2
-
b2
[a
cosh
bx
-
b
sinh
bx]
e2ax x
+
4a 2
a=b a=b
sinh axdx = 1 cosh ax a
(111) (112)
eax sinh bxdx =
eax
a2
-
b2
[-b
cosh
bx
+
a
sinh
bx]
e2ax x
-
4a 2
a=b a=b
(113)
eax tanh bxdx =
e(a+2b)x
(a
+
2b)
2
F1
1+
a , 1, 2 +
a , -e2bx
2b
2b
-
1 a
eax
2
F1
a , 1, 1E, -e2bx 2b
eax
-
2
tan-1[eax]
a
a = b (114) a=b
1 tanh ax dx = ln cosh ax
a
(115)
1 cos ax cosh bxdx = a2 + b2 [a sin ax cosh bx
+b cos ax sinh bx]
(116)
1 cos ax sinh bxdx = a2 + b2 [b cos ax cosh bx+
a sin ax sinh bx]
(117)
1
sin ax cosh bxdx =
[-a cos ax cosh bx+
a2 + b2
b sin ax sinh bx]
(118)
1
sin ax sinh bxdx =
[b cosh bx sin ax-
a2 + b2
a cos ax sinh bx]
(119)
1 sinh ax cosh axdx = [-2ax + sinh 2ax]
4a
(120)
1 sinh ax cosh bxdx = b2 - a2 [b cosh bx sinh ax
-a cosh ax sinh bx]
(121)
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