Table of Basic Integrals Basic Forms
Table of Basic Integrals
Basic Forms
(1)
xndx = 1 xn+1, n = -1
n+1
1
(2)
dx = ln |x|
x
(3)
udv = uv - vdu
1
1
(4)
dx = ln |ax + b|
ax + b a
Integrals of Rational Functions
1
1
(5)
dx = -
(x + a)2
x+a
(6)
(x
+ a)ndx
=
(x
+
a)n+1 ,n
=
-1
n+1
(7)
x(x + a)ndx = (x + a)n+1((n + 1)x - a)
(n + 1)(n + 2)
(8)
1
1 +
x2
dx
=
tan-1
x
(9)
1 dx = 1 tan-1 x
a2 + x2
a
a
1
(10)
x dx = 1 ln |a2 + x2|
a2 + x2
2
(11)
a2
x2 +
x2
dx
=
x
-
a
tan-1
x a
(12)
x3 dx = 1 x2 - 1 a2 ln |a2 + x2|
a2 + x2
22
(13)
ax2
1 + bx +
dx c
=
2
4ac - b2
tan-1
2ax + b
4ac - b2
1
1 a+x
(14)
dx =
ln
, a=b
(x + a)(x + b) b - a b + x
x
a
(15)
(x + a)2 dx = a + x + ln |a + x|
(16)
x
dx = 1 ln |ax2+bx+c|- b
tan-1
2ax
+
b
ax2 + bx + c
2a
a 4ac - b2
4ac - b2
Integrals with Roots
(17)
x
-
a
dx
=
2 (x
-
a)3/2
3
1
(18)
dx = 2 x ? a
x?a
(19)
1
dx = -2 a - x
a-x
2
2a 3
(x
-
a)3/2
+
2 5
(x
-
a)5/2
,
or
(20)
x x - a dx =
2 3
x(x
-
a)3/2
-
4 15
(x
-
a)5/2
,
or
2 15
(2a
+
3x)(x
-
a)3/2
2b 2x
(21)
ax + b dx = +
ax + b
3a 3
(22)
(ax + b)3/2 dx = 2 (ax + b)5/2
5a
x
2
(23)
dx = (x 2a) x ? a
x?a
3
(24)
x dx = - x(a - x) - a tan-1 x(a - x)
a-x
x-a
x
(25)
dx = x(a + x) - a ln x + x + a
a+x
(26)
x ax
+
b
dx
=
2 15a2
(-2b2
+
abx
+
3a2x2) ax
+
b
(27)
1 x(ax + b) dx =
(2ax + b)
ax(ax + b) - b2 ln
a x+
a(ax + b)
4a3/2
(28)
b
b2 x
x3(ax + b) dx = 12a - 8a2x + 3
b3
x3(ax + b)+ 8a5/2 ln a x + a(ax + b)
(29)
x2 ? a2
dx =
1
x x2
?
a2
?
1 a2 ln
x + x2 ? a2
2
2
3
(30)
a2
-
x2
dx
=
1
x a2
-
x2
+
1 a2
tan-1
x
2
2
a2 - x2
(31)
1
x x2 ? a2 dx =
x2 ? a2 3/2
3
(32)
1
dx = ln x + x2 ? a2
x2 ? a2
(33)
1
dx = sin-1 x
a2 - x2
a
x
(34)
dx = x2 ? a2
x2 ? a2
(35)
x
dx = - a2 - x2
a2 - x2
(36)
x2
dx =
1
x x2
?
a2
1 a2 ln
x + x2 ? a2
x2 ? a2
2
2
(37)
ax2 + bx + c dx =
b
+
2ax
ax2
+
bx
+
4ac c+
-
b2
ln
2ax + b + 2
a(ax2 + bx+c)
4a
8a3/2
(38)
1
x ax2 + bx + c dx = 48a5/2
2 a ax2 + bx + c
-3b2 + 2abx + 8a(c + ax2)
+3(b3 - 4abc) ln
b + 2ax + 2 a ax2 + bx + c
4
(39)
1
1 dx = ln 2ax + b + 2
a(ax2 + bx + c)
ax2 + bx + c
a
(40)
x
1
b
dx =
ax2 + bx + c
a
ax2 + bx + c- 2a3/2 ln 2ax + b + 2
a(ax2 + bx + c)
dx
x
(41)
(a2
+ x2)3/2
=
a2 a2
+ x2
Integrals with Logarithms
(42)
ln ax dx = x ln ax - x
(43)
x ln x dx = 1 x2 ln x - x2
2
4
(44)
x2 ln x dx = 1 x3 ln x - x3
3
9
(45)
xn ln x dx = xn+1
ln x
1
n + 1 - (n + 1)2
,
n = -1
(46)
ln ax dx = 1 (ln ax)2
x
2
ln x
1 ln x
(47)
x2
dx = - - x
x
5
b
(48)
ln(ax + b) dx = x + ln(ax + b) - x, a = 0
a
(49)
ln(x2 + a2) dx = x ln(x2 + a2) + 2a tan-1 x - 2x
a
(50)
ln(x2 - a2) dx = x ln(x2 - a2) + a ln x + a - 2x
x-a
(51)
ln ax2 + bx + c
dx
=
1 4ac
-
b2 tan-1
2ax
+b
-2x+
b +x
ln ax2 + bx + c
a
4ac - b2
2a
(52)
x ln(ax + b) dx = bx - 1 x2 + 1 x2 - b2 ln(ax + b)
2a 4 2
a2
(53)
x ln a2 - b2x2 dx = - 1 x2 + 1 x2 - a2 ln a2 - b2x2
22
b2
(54)
(ln x)2 dx = 2x - 2x ln x + x(ln x)2
(55)
(ln x)3 dx = -6x + x(ln x)3 - 3x(ln x)2 + 6x ln x
(56)
x(ln x)2 dx = x2 + 1 x2(ln x)2 - 1 x2 ln x
42
2
(57)
x2(ln x)2 dx = 2x3 + 1 x3(ln x)2 - 2 x3 ln x
27 3
9
6
Integrals with Exponentials
(58)
eax dx = 1 eax
a
(59) xeax dx = 1 xeax +
i
erf
i ax
,
where erf(x) = 2
x
e-t2 dt
a
2a3/2
0
(60)
xex dx = (x - 1)ex
(61)
xeax dx =
x1 -
eax
a a2
(62)
x2ex dx = x2 - 2x + 2 ex
(63)
x2eax dx =
x2 2x 2 a - a2 + a3
eax
(64)
x3ex dx = x3 - 3x2 + 6x - 6 ex
(65)
xneax dx = xneax - n aa
xn-1eax dx
(66)
xneax
dx
=
(-1)n an+1 [1 + n, -ax],
where
(a, x) =
ta-1e-t dt
x
(67)
eax2 dx = - i erf
ix a
2a
7
(68)
e-ax2 dx =
erf x a
2a
(69)
xe-ax2 dx = - 1 e-ax2 2a
(70)
x2e-ax2 dx = 1
erf(x a) -
x e-ax2
4 a3
2a
Integrals with Trigonometric Functions
1
(71)
sin ax dx = - cos ax
a
(72)
sin2 ax dx = x - sin 2ax
2 4a
(73)
sin3
ax
dx
=
3 -
cos
ax
+
cos
3ax
4a
12a
(74)
sinn ax
dx
1 = - cos ax
a
2F1
1 , 1 - n , 3 , cos2 ax 222
1
(75)
cos ax dx = sin ax
a
(76)
cos2 ax dx = x + sin 2ax
2 4a
(77)
cos3 axdx = 3 sin ax + sin 3ax
4a
12a
8
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