Table of Integrals

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Contents



Basic Forms

(1) $\displaystyle \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1$

(2) $\displaystyle \int \frac{1}{x}dx = \ln \vert x\vert$

(3) $\displaystyle \int u dv = uv - \int v du$

(4) $\displaystyle \int \frac{1}{ax+b}dx = \frac{1}{a} \ln \vert ax + b\vert$

Integrals of Rational Functions

(5) $\displaystyle \int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a}$

(6) $\displaystyle \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1$

(7) $\displaystyle \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)}$

(8) $\displaystyle \int \frac{1}{1+x^2}dx = \tan^{-1}x$

(9) $\displaystyle \int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a}$

(10) $\displaystyle \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln\vert a^2+x^2\vert$

(11) $\displaystyle \int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a}$

(12) $\displaystyle \int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln\vert a^2+x^2\vert$

(13) $\displaystyle \int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$

(14) $\displaystyle \int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x},$    $\displaystyle a\ne b$

(15) $\displaystyle \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln \vert a+x\vert$

(16) $\displaystyle \int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln\vert ax^2+bx+c\vert -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$

Integrals with Roots

(17) $\displaystyle \int \sqrt{x-a} dx = \frac{2}{3}(x-a)^{3/2}$

(18) $\displaystyle \int \frac{1}{\sqrt{x\pm a}} dx = 2\sqrt{x\pm a}$

(19) $\displaystyle \int \frac{1}{\sqrt{a-x}} dx = -2\sqrt{a-x}$

(20) $\displaystyle \int x\sqrt{x-a} dx = \left\{ \begin{array}{l} \frac{2 a}{3} \le...
...} (x-a)^{5/2}, \text{ or}  \frac{2}{15}(2a+3x)(x-a)^{3/2} \end{array} \right.$

(21) $\displaystyle \int \sqrt{ax+b} dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}$

(22) $\displaystyle \int (ax+b)^{3/2} dx =\frac{2}{5a}(ax+b)^{5/2}$

(23) $\displaystyle \int \frac{x}{\sqrt{x\pm a} }  dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a}$

(24) $\displaystyle \int \sqrt{\frac{x}{a-x}} dx = -\sqrt{x(a-x)} -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a}$

(25) $\displaystyle \int \sqrt{\frac{x}{a+x}} dx = \sqrt{x(a+x)} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right]$

(26) $\displaystyle \int x \sqrt{ax + b} dx = \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \sqrt{ax+b}$

(27) $\displaystyle \int \sqrt{x(ax+b)} dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} -b^2 \ln \left\vert a\sqrt{x} + \sqrt{a(ax+b)} \right\vert \right ]$

(28) $\displaystyle \int \sqrt{x^3(ax+b)}  dx =\left [ \frac{b}{12a}- \frac{b^2}{8a^...
...} + \frac{b^3}{8a^{5/2}}\ln \left \vert a\sqrt{x} + \sqrt{a(ax+b)} \right \vert$

(29) $\displaystyle \int\sqrt{x^2 \pm a^2} dx = \frac{1}{2}x\sqrt{x^2\pm a^2} \pm\frac{1}{2}a^2 \ln \left \vert x + \sqrt{x^2\pm a^2} \right \vert$

(30) $\displaystyle \int \sqrt{a^2 - x^2} dx = \frac{1}{2} x \sqrt{a^2-x^2} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}$

(31) $\displaystyle \int x \sqrt{x^2 \pm a^2} dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2}$

(32) $\displaystyle \int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln \left \vert x + \sqrt{x^2 \pm a^2} \right \vert$

(33) $\displaystyle \int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\frac{x}{a}$

(34) $\displaystyle \int \frac{x}{\sqrt{x^2\pm a^2}} dx = \sqrt{x^2 \pm a^2}$

(35) $\displaystyle \int \frac{x}{\sqrt{a^2-x^2}} dx = -\sqrt{a^2-x^2}$

(36) $\displaystyle \int \frac{x^2}{\sqrt{x^2 \pm a^2}} dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \mp \frac{1}{2}a^2 \ln \left\vert x + \sqrt{x^2\pm a^2} \right \vert$

(37) $\displaystyle \int \sqrt{a x^2 + b x + c} dx = \frac{b+2ax}{4a}\sqrt{ax^2+bx+c...
...ac{4ac-b^2}{8a^{3/2}}\ln \left\vert 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right \vert$

(38) \begin{displaymath}\begin{split}\int &x \sqrt{a x^2 + bx + c} dx = \frac{1}{48a...
...2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right\vert \right) \end{split}\end{displaymath}

(39) $\displaystyle \int\frac{1}{\sqrt{ax^2+bx+c}} dx= \frac{1}{\sqrt{a}}\ln \left\vert 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right \vert$

(40) $\displaystyle \int \frac{x}{\sqrt{ax^2+bx+c}} dx= \frac{1}{a}\sqrt{ax^2+bx + c} - \frac{b}{2a^{3/2}}\ln \left\vert 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right \vert$

(41) $\displaystyle \int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}}$

Integrals with Logarithms

(42) $\displaystyle \int \ln ax dx = x \ln ax - x$

(43) $\displaystyle \int x \ln x  dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4}$

(44) $\displaystyle \int x^2 \ln x  dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9}$

(45) $\displaystyle \int x^n \ln x dx = x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex} n\neq -1$

(46) $\displaystyle \int \frac{\ln ax}{x} dx = \frac{1}{2}\left ( \ln ax \right)^2$

(47) $\displaystyle \int \frac{\ln x}{x^2} dx = -\frac{1}{x}-\frac{\ln x}{x}$

(48) $\displaystyle \int \ln (ax + b)  dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) - x , a\ne 0$

(49) $\displaystyle \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} - 2x$

(50) $\displaystyle \int \ln ( x^2 - a^2 )\hspace{.5ex} {dx} = x \ln (x^2 - a^2 ) +a\ln \frac{x+a}{x-a} - 2x$

(51) $\displaystyle \int \ln \left ( ax^2 + bx + c\right)  dx = \frac{1}{a}\sqrt{4ac...
...sqrt{4ac-b^2}} -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right)$

(52) $\displaystyle \int x \ln (ax + b) dx = \frac{bx}{2a}-\frac{1}{4}x^2 +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b)$

(53) $\displaystyle \int x \ln \left ( a^2 - b^2 x^2 \right ) dx = -\frac{1}{2}x^2+ \frac{1}{2}\left( x^2 - \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right)$

(54) $\displaystyle \int (\ln x)^2 dx = 2x - 2x \ln x + x (\ln x)^2$

(55) $\displaystyle \int (\ln x)^3 dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x$

(56) $\displaystyle \int x (\ln x)^2 dx = \frac{x^2}{4}+\frac{1}{2} x^2 (\ln x)^2-\frac{1}{2} x^2 \ln x$

(57) $\displaystyle \int x^2 (\ln x)^2 dx = \frac{2 x^3}{27}+\frac{1}{3} x^3 (\ln x)^2-\frac{2}{9} x^3 \ln x$

Integrals with Exponentials

(58) $\displaystyle \int e^{ax} dx = \frac{1}{a}e^{ax}$

(59) $\displaystyle \int \sqrt{x} e^{ax} dx = \frac{1}{a}\sqrt{x}e^{ax} +\frac{i\sqrt{\pi}}{2a^{3/2}}$   erf$\displaystyle \left(i\sqrt{ax}\right),$    where erf$\displaystyle (x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$

(60) $\displaystyle \int x e^x dx = (x-1) e^x$

(61) $\displaystyle \int x e^{ax} dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax}$

(62) $\displaystyle \int x^2 e^{x} dx = \left(x^2 - 2x + 2\right) e^{x}$

(63) $\displaystyle \int x^2 e^{ax} dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax}$

(64) $\displaystyle \int x^3 e^{x} dx = \left(x^3-3x^2 + 6x - 6\right) e^{x}$

(65) $\displaystyle \int x^n e^{ax} dx = \dfrac{x^n e^{ax}}{a} - \dfrac{n}{a}\int x^{n-1}e^{ax}$   d$\displaystyle x$

(66) $\displaystyle \int x^n e^{ax} dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax],$    where $\displaystyle \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}$   d$\displaystyle t$

(67) $\displaystyle \int e^{ax^2} dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}$erf$\displaystyle \left(ix\sqrt{a}\right)$

(68) $\displaystyle \int e^{-ax^2} dx = \frac{\sqrt{\pi}}{2\sqrt{a}}$erf$\displaystyle \left(x\sqrt{a}\right)$

(69) $\displaystyle \int x e^{-ax^2} {dx} = -\dfrac{1}{2a}e^{-ax^2}$

(70) $\displaystyle \int x^2 e^{-ax^2} {dx} = \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}$erf$\displaystyle (x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2}$

Integrals with Trigonometric Functions

(71) $\displaystyle \int \sin ax  dx = -\frac{1}{a} \cos ax$

(72) $\displaystyle \int \sin^2 ax dx = \frac{x}{2} - \frac{\sin 2ax} {4a}$

(73) $\displaystyle \int \sin^3 ax  dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a}$

(74) $\displaystyle \int \sin^n ax  dx = -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \right]$

(75) $\displaystyle \int \cos ax dx= \frac{1}{a} \sin ax$

(76) $\displaystyle \int \cos^2 ax dx = \frac{x}{2}+\frac{ \sin 2ax}{4a}$

(77) $\displaystyle \int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a}$

(78) $\displaystyle \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \right]$

(79) $\displaystyle \int \cos x \sin x dx = \frac{1}{2}\sin^2 x + c_1 = -\frac{1}{2} \cos^2x + c_2 = -\frac{1}{4} \cos 2x + c_3$

(80) $\displaystyle \int \cos ax \sin bx dx = \frac{\cos[(a-b) x]}{2(a-b)} - \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b$

(81) $\displaystyle \int \sin^2 ax \cos bx dx = -\frac{\sin[(2a-b)x]}{4(2a-b)} + \frac{\sin bx}{2b} - \frac{\sin[(2a+b)x]}{4(2a+b)}$

(82) $\displaystyle \int \sin^2 x \cos x dx = \frac{1}{3} \sin^3 x$

(83) $\displaystyle \int \cos^2 ax \sin bx dx = \frac{\cos[(2a-b)x]}{4(2a-b)} - \frac{\cos bx}{2b} - \frac{\cos[(2a+b)x]}{4(2a+b)}$

(84) $\displaystyle \int \cos^2 ax \sin ax dx = -\frac{1}{3a}\cos^3{ax}$

(85) $\displaystyle \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} -\frac{\sin 2ax}{8a}- \frac{\sin[2(a-b)x]}{16(a-b)} +\frac{\sin 2bx}{8b}- \frac{\sin[2(a+b)x]}{16(a+b)}$

(86) $\displaystyle \int \sin^2 ax \cos^2 ax dx = \frac{x}{8}-\frac{\sin 4ax}{32a}$

(87) $\displaystyle \int \tan ax dx = -\frac{1}{a} \ln \cos ax$

(88) $\displaystyle \int \tan^2 ax dx = -x + \frac{1}{a} \tan ax$

(89) $\displaystyle \int \tan^n ax dx = \frac{\tan^{n+1} ax }{a(1+n)} \times {_2}F_1\left( \frac{n+1}{2}, 1, \frac{n+3}{2}, -\tan^2 ax \right)$

(90) $\displaystyle \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax$

(91) $\displaystyle \int \sec x  dx = \ln \vert \sec x + \tan x \vert = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right)$

(92) $\displaystyle \int \sec^2 ax dx = \frac{1}{a} \tan ax$

(93) $\displaystyle \int \sec^3 x  {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln \vert \sec x + \tan x \vert$

(94) $\displaystyle \int \sec x \tan x dx = \sec x$

(95) $\displaystyle \int \sec^2 x \tan x dx = \frac{1}{2} \sec^2 x$

(96) $\displaystyle \int \sec^n x \tan x  dx = \frac{1}{n} \sec^n x , n\ne 0$

(97) $\displaystyle \int \csc x dx = \ln \left \vert \tan \frac{x}{2} \right\vert = \ln \vert \csc x - \cot x\vert + C$

(98) $\displaystyle \int \csc^2 ax dx = -\frac{1}{a} \cot ax$

(99) $\displaystyle \int \csc^3 x dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln \vert \csc x - \cot x \vert$

(100) $\displaystyle \int \csc^nx \cot x dx = -\frac{1}{n}\csc^n x, n\ne 0$

(101) $\displaystyle \int \sec x \csc x  dx = \ln \vert \tan x \vert$

Products of Trigonometric Functions and Monomials

(102) $\displaystyle \int x \cos x  dx = \cos x + x \sin x$

(103) $\displaystyle \int x \cos ax  dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax$

(104) $\displaystyle \int x^2 \cos x  dx = 2 x \cos x + \left ( x^2 - 2 \right ) \sin x$

(105) $\displaystyle \int x^2 \cos ax  dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 - 2 }{a^3} \sin ax$

(106) $\displaystyle \int x^n \cos x dx = -\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) + (-1)^n \Gamma(n+1, ix)\right]$

(107) $\displaystyle \int x^n \cos ax  dx = \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) -\Gamma(n+1, ixa)\right]$

(108) $\displaystyle \int x \sin x dx = -x \cos x + \sin x$

(109) $\displaystyle \int x \sin ax dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2}$

(110) $\displaystyle \int x^2 \sin x dx = \left(2-x^2\right) \cos x + 2 x \sin x$

(111) $\displaystyle \int x^2 \sin ax dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2}$

(112) $\displaystyle \int x^n \sin x  dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) - (-1)^n\Gamma(n+1, -ix)\right]$

(113) $\displaystyle \int x \cos^2 x  dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x$

(114) $\displaystyle \int x \sin^2 x  dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x - \frac{1}{4} x \sin 2x$

(115) $\displaystyle \int x \tan^2 x  dx = -\frac{x^2}{2} + \ln \cos x + x \tan x$

(116) $\displaystyle \int x \sec^2 x  dx = \ln \cos x + x \tan x$

Products of Trigonometric Functions and Exponentials

(117) $\displaystyle \int e^x \sin x  dx = \frac{1}{2}e^x (\sin x - \cos x)$

(118) $\displaystyle \int e^{bx} \sin ax dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax - a\cos ax)$

(119) $\displaystyle \int e^x \cos x dx = \frac{1}{2}e^x (\sin x + \cos x)$

(120) $\displaystyle \int e^{bx} \cos ax dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax )$

(121) $\displaystyle \int x e^x \sin x dx = \frac{1}{2}e^x (\cos x - x \cos x + x \sin x)$

(122) $\displaystyle \int x e^x \cos x dx = \frac{1}{2}e^x (x \cos x - \sin x + x \sin x)$

Integrals of Hyperbolic Functions

(123) $\displaystyle \int \cosh ax dx =\frac{1}{a} \sinh ax$

(124) $\displaystyle \int e^{ax} \cosh bx  dx = \begin{cases}\displaystyle{\frac{e^{a...
...& a\ne b  \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{cases}$

(125) $\displaystyle \int \sinh ax dx = \frac{1}{a} \cosh ax$

(126) $\displaystyle \int e^{ax} \sinh bx  dx = \begin{cases}\displaystyle{\frac{e^{a...
...& a\ne b  \displaystyle{\frac{e^{2ax}}{4a} - \frac{x}{2}} & a = b \end{cases}$

(127) $\displaystyle \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax$

(128) $\displaystyle \int e^{ax} \tanh bx dx = \begin{cases}\displaystyle{ \frac{ e^{...
...ne b  \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b \end{cases}$

(129) $\displaystyle \int \cos ax \cosh bx dx = \frac{1}{a^2 + b^2} \left[ a \sin ax \cosh bx + b \cos ax \sinh bx \right]$

(130) $\displaystyle \int \cos ax \sinh bx dx = \frac{1}{a^2 + b^2} \left[ b \cos ax \cosh bx + a \sin ax \sinh bx \right]$

(131) $\displaystyle \int \sin ax \cosh bx  dx = \frac{1}{a^2 + b^2} \left[ -a \cos ax \cosh bx + b \sin ax \sinh bx \right]$

(132) $\displaystyle \int \sin ax \sinh bx  dx = \frac{1}{a^2 + b^2} \left[ b \cosh bx \sin ax - a \cos ax \sinh bx \right]$

(133) $\displaystyle \int \sinh ax \cosh ax dx= \frac{1}{4a}\left[ -2ax + \sinh 2ax \right]$

(134) $\displaystyle \int \sinh ax \cosh bx  dx = \frac{1}{b^2-a^2}\left[ b \cosh bx \sinh ax - a \cosh ax \sinh bx \right]$


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Acknowledgements

The author is not in any way affiliated with Wolfram Research, Mathematica, or the Wolfram Integrater. I've just posted the link at the top of this page because I think their web site is really cool!

Thanks to the following for pointing out corrections, typos, or suggesting new formulas: LS Rigo (19); Stephen Gilmore (20); Bruce Weems (23); Stephen Russ (36); Justin Winokur (37), (41); Larry Morris and Tom Gatliffe (38); James Duley (40); Daniel Ajoy (59); Johannes Ebke (66); Jim Swift (65),(66), (67), (68); Kregg Quarles (69),(70); Vedran (Veky) Čačić (79); Peter Kloeppel (93); Phillipe (Xul) (112); Nicole Ritzert and Andrea Bajo (118); Jose Antonio Alvarez Loyo Yates (127); Corné de Witt (128).

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