Integral Table 2/2

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Integrator

Table of Basic Integrals

Basic Forms

\int x^{n} d x = \frac{1}{n + 1} x^{n + 1}, n \neq - 1

(1)

\int \frac{1}{x} d x = ln | x |

(2)

\int u d v = u v - \int v d u

(3)

\int \frac{1}{a x + b} d x = \frac{1}{a} ln | a x + b |

(4)

Integrals of Rational Functions

\int \frac{1}{{(x + a)}^{2}} d x = - \frac{1}{x + a}

(5)

\int {(x + a)}^{n} d x = \frac{{(x + a)}^{n + 1}}{n + 1}, n \neq - 1

(6)

\int x {(x + a)}^{n} d x = \frac{{(x + a)}^{n + 1} ((n + 1) x - a)}{(n + 1) (n + 2)}

(7)

\int \frac{1}{1 + x^{2}} d x = {tan}^{- 1} x

(8)

\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} {tan}^{- 1} \frac{x}{a}

(9)

\int \frac{x}{a^{2} + x^{2}} d x = \frac{1}{2} ln | a^{2} + x^{2} |

(10)

\int \frac{x^{2}}{a^{2} + x^{2}} d x = x - a {tan}^{- 1} \frac{x}{a}

(11)

\int \frac{x^{3}}{a^{2} + x^{2}} d x = \frac{1}{2} x^{2} - \frac{1}{2} a^{2} ln | a^{2} + x^{2} |

(12)

\int \frac{1}{a x^{2} + b x + c} d x = \frac{2}{\sqrt{4 a c - b^{2}}} {tan}^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^{2}}}

(13)

\int \frac{1}{(x + a) (x + b)} d x = \frac{1}{b - a} ln \frac{a + x}{b + x}, a \neq b

(14)

\int \frac{x}{{(x + a)}^{2}} d x = \frac{a}{a + x} + ln | a + x |

(15)

\int \frac{x}{a x^{2} + b x + c} d x = \frac{1}{2 a} ln | a x^{2} + b x + c | - \frac{b}{a \sqrt{4 a c - b^{2}}} {tan}^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^{2}}}

(16)

Integrals with Roots

\int \sqrt{x - a} d x = \frac{2}{3} {(x - a)}^{3 ∕ 2}

(17)

\int \frac{1}{\sqrt{x \pm a}} d x = 2 \sqrt{x \pm a}

(18)

\int \frac{1}{\sqrt{a - x}} d x = - 2 \sqrt{a - x}

(19)

\int x \sqrt{x - a} d x = \{\begin{matrix} \frac{2 a}{3} {(x - a)}^{3 ∕ 2} + \frac{2}{5} {(x - a)}^{5 ∕ 2}, or \\ \frac{2}{3} x {(x - a)}^{3 ∕ 2} - \frac{4}{15} {(x - a)}^{5 ∕ 2}, or \\ \frac{2}{15} (2 a + 3 x) {(x - a)}^{3 ∕ 2} \end{matrix}

(20)

\int \sqrt{a x + b} d x = (\frac{2 b}{3 a} + \frac{2 x}{3}) \sqrt{a x + b}

(21)

\int {(a x + b)}^{3 ∕ 2} d x = \frac{2}{5 a} {(a x + b)}^{5 ∕ 2}

(22)

\int \frac{x}{\sqrt{x \pm a}} d x = \frac{2}{3} (x \mp 2 a) \sqrt{x \pm a}

(23)

\int \sqrt{\frac{x}{a - x}} d x = - \sqrt{x (a - x)} - a {tan}^{- 1} \frac{\sqrt{x (a - x)}}{x - a}

(24)

\int \sqrt{\frac{x}{a + x}} d x = \sqrt{x (a + x)} - a ln [\sqrt{x} + \sqrt{x + a}]

(25)

\int x \sqrt{a x + b} d x = \frac{2}{15 a^{2}} (- 2 b^{2} + a b x + 3 a^{2} x^{2}) \sqrt{a x + b}

(26)

\int \sqrt{x (a x + b)} d x = \frac{1}{4 a^{3 ∕ 2}} [(2 a x + b) \sqrt{a x (a x + b)} - b^{2} ln |a \sqrt{x} + \sqrt{a (a x + b)}|]

(27)

\int \sqrt{x^{3} (a x + b)} d x = [\frac{b}{12 a} - \frac{b^{2}}{8 a^{2} x} + \frac{x}{3}] \sqrt{x^{3} (a x + b)} + \frac{b^{3}}{8 a^{5 ∕ 2}} ln |a \sqrt{x} + \sqrt{a (a x + b)}|

(28)

\int \sqrt{x^{2} \pm a^{2}} d x = \frac{1}{2} x \sqrt{x^{2} \pm a^{2}} \pm \frac{1}{2} a^{2} ln |x + \sqrt{x^{2} \pm a^{2}}|

(29)

\int \sqrt{a^{2} - x^{2}} d x = \frac{1}{2} x \sqrt{a^{2} - x^{2}} + \frac{1}{2} a^{2} {tan}^{- 1} \frac{x}{\sqrt{a^{2} - x^{2}}}

(30)

\int x \sqrt{x^{2} \pm a^{2}} d x = \frac{1}{3} {(x^{2} \pm a^{2})}^{3 ∕ 2}

(31)

\int \frac{1}{\sqrt{x^{2} \pm a^{2}}} d x = ln |x + \sqrt{x^{2} \pm a^{2}}|

(32)

\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = {sin}^{- 1} \frac{x}{a}

(33)

\int \frac{x}{\sqrt{x^{2} \pm a^{2}}} d x = \sqrt{x^{2} \pm a^{2}}

(34)

\int \frac{x}{\sqrt{a^{2} - x^{2}}} d x = - \sqrt{a^{2} - x^{2}}

(35)

\int \frac{x^{2}}{\sqrt{x^{2} \pm a^{2}}} d x = \frac{1}{2} x \sqrt{x^{2} \pm a^{2}} \mp \frac{1}{2} a^{2} ln |x + \sqrt{x^{2} \pm a^{2}}|

(36)

\int \sqrt{a x^{2} + b x + c} d x = \frac{b + 2 a x}{4 a} \sqrt{a x^{2} + b x + c} + \frac{4 a c - b^{2}}{8 a^{3 ∕ 2}} ln |2 a x + b + 2 \sqrt{a (a x^{2} + b x^{+} c)}|

(37)

\begin{matrix} \begin{aligned} \int & x \sqrt{a x^{2} + b x + c} d x = \frac{1}{48 a^{5 ∕ 2}} (2 \sqrt{a} \sqrt{a x^{2} + b x + c} (- 3 b^{2} + 2 a b x + 8 a (c + a x^{2})) \\ + 3 (b^{3} - 4 a b c) ln |b + 2 a x + 2 \sqrt{a} \sqrt{a x^{2} + b x + c}|) \end{aligned} \end{matrix}

(38)

\int \frac{1}{\sqrt{a x^{2} + b x + c}} d x = \frac{1}{\sqrt{a}} ln |2 a x + b + 2 \sqrt{a (a x^{2} + b x + c)}|

(39)

\int \frac{x}{\sqrt{a x^{2} + b x + c}} d x = \frac{1}{a} \sqrt{a x^{2} + b x + c} - \frac{b}{2 a^{3 ∕ 2}} ln |2 a x + b + 2 \sqrt{a (a x^{2} + b x + c)}|

(40)

\int \frac{d x}{{(a^{2} + x^{2})}^{3 ∕ 2}} = \frac{x}{a^{2} \sqrt{a^{2} + x^{2}}}

(41)

Integrals with Logarithms

\int ln a x d x = x ln a x - x

(42)

\int x ln x d x = \frac{1}{2} x^{2} ln x - \frac{x^{2}}{4}

(43)

\int x^{2} ln x d x = \frac{1}{3} x^{3} ln x - \frac{x^{3}}{9}

(44)

\int x^{n} ln x d x = x^{n + 1} (\frac{ln x}{n + 1} - \frac{1}{{(n + 1)}^{2}}), n \neq - 1

(45)

\int \frac{ln a x}{x} d x = \frac{1}{2} {(ln a x)}^{2}

(46)

\int \frac{ln x}{x^{2}} d x = - \frac{1}{x} - \frac{ln x}{x}

(47)

\int ln (a x + b) d x = (x + \frac{b}{a}) ln (a x + b) - x, a \neq 0

(48)

\int ln (x^{2} + a^{2}) d x = x ln (x^{2} + a^{2}) + 2 a {tan}^{- 1} \frac{x}{a} - 2 x

(49)

\int ln (x^{2} - a^{2}) d x = x ln (x^{2} - a^{2}) + a ln \frac{x + a}{x - a} - 2 x

(50)

\int ln (a x^{2} + b x + c) d x = \frac{1}{a} \sqrt{4 a c - b^{2}} {tan}^{- 1} \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} - 2 x + (\frac{b}{2 a} + x) ln (a x^{2} + b x + c)

(51)

\int x ln (a x + b) d x = \frac{b x}{2 a} - \frac{1}{4} x^{2} + \frac{1}{2} (x^{2} - \frac{b^{2}}{a^{2}}) ln (a x + b)

(52)

\int x ln (a^{2} - b^{2} x^{2}) d x = - \frac{1}{2} x^{2} + \frac{1}{2} (x^{2} - \frac{a^{2}}{b^{2}}) ln (a^{2} - b^{2} x^{2})

(53)

\int {(ln x)}^{2} d x = 2 x - 2 x ln x + x {(ln x)}^{2}

(54)

\int {(ln x)}^{3} d x = - 6 x + x {(ln x)}^{3} - 3 x {(ln x)}^{2} + 6 x ln x

(55)

\int x {(ln x)}^{2} d x = \frac{x^{2}}{4} + \frac{1}{2} x^{2} {(ln x)}^{2} - \frac{1}{2} x^{2} ln x

(56)

\int x^{2} {(ln x)}^{2} d x = \frac{2 x^{3}}{27} + \frac{1}{3} x^{3} {(ln x)}^{2} - \frac{2}{9} x^{3} ln x

(57)

Integrals with Exponentials

\int e^{a x} d x = \frac{1}{a} e^{a x}

(58)

\int \sqrt{x} e^{a x} d x = \frac{1}{a} \sqrt{x} e^{a x} + \frac{i \sqrt{π}}{2 a^{3 ∕ 2}} erf (i \sqrt{a x}), where erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t

(59)

\int x e^{x} d x = (x - 1) e^{x}

(60)

\int x e^{a x} d x = (\frac{x}{a} - \frac{1}{a^{2}}) e^{a x}

(61)

\int x^{2} e^{x} d x = (x^{2} - 2 x + 2) e^{x}

(62)

\int x^{2} e^{a x} d x = (\frac{x^{2}}{a} - \frac{2 x}{a^{2}} + \frac{2}{a^{3}}) e^{a x}

(63)

\int x^{3} e^{x} d x = (x^{3} - 3 x^{2} + 6 x - 6) e^{x}

(64)

\int x^{n} e^{a x} d x = \frac{x^{n} e^{a x}}{a} - \frac{n}{a} \int x^{n - 1} e^{a x} d x

(65)

\int x^{n} e^{a x} d x = \frac{{(- 1)}^{n}}{a^{n + 1}} Γ [1 + n, - a x], where Γ (a, x) = \int_{x}^{\infty} t^{a - 1} e^{- t} d t

(66)

\int e^{a x^{2}} d x = - \frac{i \sqrt{π}}{2 \sqrt{a}} erf (i x \sqrt{a})

(67)

\int e^{- a x^{2}} d x = \frac{\sqrt{π}}{2 \sqrt{a}} erf (x \sqrt{a})

(68)

\int x e^{- a x^{2}} d x = - \frac{1}{2 a} e^{- a x^{2}}

(69)

\int x^{2} e^{- a x^{2}} d x = \frac{1}{4} \sqrt{\frac{π}{a^{3}}} erf (x \sqrt{a}) - \frac{x}{2 a} e^{- a x^{2}}

(70)

Integrals with Trigonometric Functions

\int sin a x d x = - \frac{1}{a} cos a x

(71)

\int {sin}^{2} a x d x = \frac{x}{2} - \frac{sin 2 a x}{4 a}

(72)

\int {sin}^{3} a x d x = - \frac{3 cos a x}{4 a} + \frac{cos 3 a x}{12 a}

(73)

\int {sin}^{n} a x d x = - \frac{1}{a} cos a x_{2} F_{1} [\frac{1}{2}, \frac{1 - n}{2}, \frac{3}{2}, {cos}^{2} a x]

(74)

\int cos a x d x = \frac{1}{a} sin a x

(75)

\int {cos}^{2} a x d x = \frac{x}{2} + \frac{sin 2 a x}{4 a}

(76)

\int {cos}^{3} a x d x = \frac{3 sin a x}{4 a} + \frac{sin 3 a x}{12 a}

(77)

\int {cos}^{p} a x d x = - \frac{1}{a (1 + p)} {cos}^{1 + p} a x \times_{2} F_{1} [\frac{1 + p}{2}, \frac{1}{2}, \frac{3 + p}{2}, {cos}^{2} a x]

(78)

\int cos x sin x d x = \frac{1}{2} {sin}^{2} x + c_{1} = - \frac{1}{2} {cos}^{2} x + c_{2} = - \frac{1}{4} cos 2 x + c_{3}

(79)

\int cos a x sin b x d x = \frac{cos [(a - b) x]}{2 (a - b)} - \frac{cos [(a + b) x]}{2 (a + b)}, a \neq b

(80)

\int {sin}^{2} a x cos b x d x = - \frac{sin [(2 a - b) x]}{4 (2 a - b)} + \frac{sin b x}{2 b} - \frac{sin [(2 a + b) x]}{4 (2 a + b)}

(81)

\int {sin}^{2} x cos x d x = \frac{1}{3} {sin}^{3} x

(82)

\int {cos}^{2} a x sin b x d x = \frac{cos [(2 a - b) x]}{4 (2 a - b)} - \frac{cos b x}{2 b} - \frac{cos [(2 a + b) x]}{4 (2 a + b)}

(83)

\int {cos}^{2} a x sin a x d x = - \frac{1}{3 a} {cos}^{3} a x

(84)

\int {sin}^{2} a x {cos}^{2} b x d x = \frac{x}{4} - \frac{sin 2 a x}{8 a} - \frac{sin [2 (a - b) x]}{16 (a - b)} + \frac{sin 2 b x}{8 b} - \frac{sin [2 (a + b) x]}{16 (a + b)}

(85)

\int {sin}^{2} a x {cos}^{2} a x d x = \frac{x}{8} - \frac{sin 4 a x}{32 a}

(86)

\int tan a x d x = - \frac{1}{a} ln cos a x

(87)

\int {tan}^{2} a x d x = - x + \frac{1}{a} tan a x

(88)

\int {tan}^{n} a x d x = \frac{{tan}^{n + 1} a x}{a (1 + n)} \times_{2} F_{1} (\frac{n + 1}{2}, 1, \frac{n + 3}{2}, - {tan}^{2} a x)

(89)

\int {tan}^{3} a x d x = \frac{1}{a} ln cos a x + \frac{1}{2 a} {sec}^{2} a x

(90)

\int sec x d x = ln | sec x + tan x | = 2 {tanh}^{- 1} (tan \frac{x}{2})

(91)

\int {sec}^{2} a x d x = \frac{1}{a} tan a x

(92)

\int {sec}^{3} x d x = \frac{1}{2} sec x tan x + \frac{1}{2} ln | sec x + tan x |

(93)

\int sec x tan x d x = sec x

(94)

\int {sec}^{2} x tan x d x = \frac{1}{2} {sec}^{2} x

(95)

\int {sec}^{n} x tan x d x = \frac{1}{n} {sec}^{n} x, n \neq 0

(96)

\int csc x d x = ln |tan \frac{x}{2}| = ln | csc x - cot x | + C

(97)

\int {csc}^{2} a x d x = - \frac{1}{a} cot a x

(98)

\int {csc}^{3} x d x = - \frac{1}{2} cot x csc x + \frac{1}{2} ln | csc x - cot x |

(99)

\int {csc}^{n} x cot x d x = - \frac{1}{n} {csc}^{n} x, n \neq 0

(100)

\int sec x csc x d x = ln | tan x |

(101)

Products of Trigonometric Functions and Monomials

\int x cos x d x = cos x + x sin x

(102)

\int x cos a x d x = \frac{1}{a^{2}} cos a x + \frac{x}{a} sin a x

(103)

\int x^{2} cos x d x = 2 x cos x + (x^{2} - 2) sin x

(104)

\int x^{2} cos a x d x = \frac{2 x cos a x}{a^{2}} + \frac{a^{2} x^{2} - 2}{a^{3}} sin a x

(105)

\int x^{n} cos x d x = - \frac{1}{2} {(i)}^{n + 1} [Γ (n + 1, - i x) + {(- 1)}^{n} Γ (n + 1, i x)]

(106)

\int x^{n} cos a x d x = \frac{1}{2} {(i a)}^{1 - n} [{(- 1)}^{n} Γ (n + 1, - i a x) - Γ (n + 1, i x a)]

(107)

\int x sin x d x = - x cos x + sin x

(108)

\int x sin a x d x = - \frac{x cos a x}{a} + \frac{sin a x}{a^{2}}

(109)

\int x^{2} sin x d x = (2 - x^{2}) cos x + 2 x sin x

(110)

\int x^{2} sin a x d x = \frac{2 - a^{2} x^{2}}{a^{3}} cos a x + \frac{2 x sin a x}{a^{2}}

(111)

\int x^{n} sin x d x = - \frac{1}{2} {(i)}^{n} [Γ (n + 1, - i x) - {(- 1)}^{n} Γ (n + 1, - i x)]

(112)

\int x {cos}^{2} x d x = \frac{x^{2}}{4} + \frac{1}{8} cos 2 x + \frac{1}{4} x sin 2 x

(113)

\int x {sin}^{2} x d x = \frac{x^{2}}{4} - \frac{1}{8} cos 2 x - \frac{1}{4} x sin 2 x

(114)

\int x {tan}^{2} x d x = - \frac{x^{2}}{2} + ln cos x + x tan x

(115)

\int x {sec}^{2} x d x = ln cos x + x tan x

(116)

Products of Trigonometric Functions and Exponentials

\int e^{x} sin x d x = \frac{1}{2} e^{x} (sin x - cos x)

(117)

\int e^{b x} sin a x d x = \frac{1}{a^{2} + b^{2}} e^{b x} (b sin a x - a cos a x)

(118)

\int e^{x} cos x d x = \frac{1}{2} e^{x} (sin x + cos x)

(119)

\int e^{b x} cos a x d x = \frac{1}{a^{2} + b^{2}} e^{b x} (a sin a x + b cos a x)

(120)

\int x e^{x} sin x d x = \frac{1}{2} e^{x} (cos x - x cos x + x sin x)

(121)

\int x e^{x} cos x d x = \frac{1}{2} e^{x} (x cos x - sin x + x sin x)

(122)

Integrals of Hyperbolic Functions

\int cosh a x d x = \frac{1}{a} sinh a x

(123)

\int e^{a x} cosh b x d x = \{\begin{matrix} \frac{e^{a x}}{a^{2} - b^{2}} [a cosh b x - b sinh b x] & a \neq b \\ \frac{e^{2 a x}}{4 a} + \frac{x}{2} & a = b \end{matrix}

(124)

\int sinh a x d x = \frac{1}{a} cosh a x

(125)

\int e^{a x} sinh b x d x = \{\begin{matrix} \frac{e^{a x}}{a^{2} - b^{2}} [- b cosh b x + a sinh b x] & a \neq b \\ \frac{e^{2 a x}}{4 a} - \frac{x}{2} & a = b \end{matrix}

(126)

\int tanh a x d x = \frac{1}{a} ln cosh a x

(127)

\int e^{a x} tanh b x d x = \{\begin{matrix} \frac{e^{(a + 2 b) x}}{(a + 2 b)} (_{2} F_{1}) [1 + \frac{a}{2 b}, 1, 2 + \frac{a}{2 b}, - e^{2 b x}] \\ - \frac{e^{a x}}{a} (_{2} F_{1}) [1, \frac{a}{2 b}, 1 + \frac{a}{2 b}, - e^{2 b x}] & a \neq b \\ \frac{e^{a x} - 2 {tan}^{- 1} [e^{a x}]}{a} & a = b \end{matrix}

(128)

\int cos a x cosh b x d x = \frac{1}{a^{2} + b^{2}} [a sin a x cosh b x + b cos a x sinh b x]

(129)

\int cos a x sinh b x d x = \frac{1}{a^{2} + b^{2}} [b cos a x cosh b x + a sin a x sinh b x]

(130)

\int sin a x cosh b x d x = \frac{1}{a^{2} + b^{2}} [- a cos a x cosh b x + b sin a x sinh b x]

(131)

\int sin a x sinh b x d x = \frac{1}{a^{2} + b^{2}} [b cosh b x sin a x - a cos a x sinh b x]

(132)

\int sinh a x cosh a x d x = \frac{1}{4 a} [- 2 a x + sinh 2 a x]

(133)

\int sinh a x cosh b x d x = \frac{1}{b^{2} - a^{2}} [b cosh b x sinh a x - a cosh a x sinh b x]

(134)