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Integral Table 2/2

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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)