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| {{about|mostly indefinite integrals in calculus|a list of definite integrals|List of definite integrals}}
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| {{more footnotes|date=November 2013}}
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| {{Calculus |Integral}}
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| [[Integral|Integration]] is the basic operation in [[integral calculus]]. While [[derivative|differentiation]] has easy rules by which the derivative of a complicated [[Function (mathematics)|function]] can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
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| ==Historical development of integrals==
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| A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician [[Meyer Hirsch]] in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician [[David de Bierens de Haan]]. A new edition was published in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by de Bierens are denoted by BI.
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| Not all [[closed-form expression]]s have closed-form antiderivatives; this study forms the subject of [[differential Galois theory]], which was initially developed by [[Joseph Liouville]] in the 1830s and 1840s, leading to [[Liouville's theorem (differential algebra)|Liouville's theorem]] which classifies which expressions have closed form antiderivatives. A simple example of a function without a closed form antiderivative is ''e''<sup>−''x''<sup>2</sup></sup>, whose antiderivative is (up to constants) the [[error function]].
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| Since 1968 there is the [[Risch algorithm]] for determining indefinite integrals that can be expressed in term of [[elementary function]]s, typically using a [[computer algebra system]]. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the [[Meijer G-function]].
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| ==Lists of integrals==
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| More detail may be found on the following pages for the '''lists of [[integral]]s''':
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| * [[List of integrals of rational functions]]
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| * [[List of integrals of irrational functions]]
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| * [[List of integrals of trigonometric functions]]
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| * [[List of integrals of inverse trigonometric functions]]
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| * [[List of integrals of hyperbolic functions]]
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| * [[List of integrals of inverse hyperbolic functions]]
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| * [[List of integrals of exponential functions]]
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| * [[List of integrals of logarithmic functions]]
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| * [[List of integrals of Gaussian functions]]
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| Gradshteyn, Ryzhik, Jeffrey, Zwillinger's ''Table of Integrals, Series, and Products'' contains a large collection of results. An even larger, multivolume table is the ''Integrals and Series'' by Prudnikov, Brychkov, and [[Oleg Igorevich Marichev|Marichev]] (with volumes 1–3 listing integrals and series of [[elementary function|elementary]] and [[special functions]], volume 4–5 are tables of [[Laplace transform]]s). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's ''Tables of Indefinite Integrals'', or as chapters in Zwillinger's ''CRC Standard Mathematical Tables and Formulae'', Bronstein and Semendyayev's ''Handbook of Mathematics'' (Springer) and ''Oxford Users' Guide to Mathematics'' (Oxford Univ. Press), and other mathematical handbooks.
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| Other useful resources include [[Abramowitz and Stegun]] and the [[Bateman Manuscript Project]]. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.
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| There are several web sites which have tables of integrals and integrals on demand. [[Wolfram Alpha]] can show results, and for some simpler expressions, also the intermediate steps of the integration. [[Wolfram Research]] also operates another online service, the [http://integrals.wolfram.com/index.jsp Wolfram Mathematica Online Integrator]. | |
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| ==Integrals of simple functions==
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| ''C'' is used for an [[arbitrary constant of integration]] that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.
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| These formulas only state in another form the assertions in the [[table of derivatives]].
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| ===Integrals with a singularity===
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| When there is a singularity in the function being integrated such that the integral becomes undefined, i.e., it is not [[Lebesgue integration|Lebesgue integrable]], then ''C'' does not need to be the same on both sides of the singularity. The forms below normally assume the [[Cauchy principal value]] around a singularity in the value of ''C'' but this is not in general necessary. For instance in
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| ::<math>\int {1 \over x}\,dx = \ln \left|x \right| + C</math>
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| there is a singularity at 0 and the integral becomes infinite there. If the integral above was used to give a definite integral between -1 and 1 the answer would be 0. This however is only the value assuming the Cauchy principal value for the integral around the singularity. If the integration was done in the complex plane the result would depend on the path around the origin, in this case the singularity contributes −''i''{{pi}} when using a path above the origin and ''i''{{pi}} for a path below the origin. A function on the real line could use a completely different value of ''C'' on either side of the origin as in:
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| : <math> \int {1 \over x}\,dx = \ln|x| + \begin{cases} A & \text{if }x>0; \\ B & \text{if }x < 0. \end{cases} </math>
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| ===Rational functions===
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| :''More integrals: [[List of integrals of rational functions]]''
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| These rational functions have a non-integrable singularity at 0 for ''a'' ≤ −1.
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| :<math>\int k\,dx = kx + C</math>
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| :<math>\int x^a\,dx = \frac{x^{a+1}}{a+1} + C \qquad\text{(for } a\neq -1\text{)}\,\!</math> ([[Cavalieri's quadrature formula]])
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| :<math>\int (ax + b)^n \, dx= \frac{(ax + b)^{n+1}}{a(n + 1)} + C \qquad\text{(for } n\neq -1\text{)}\,\!</math>
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| :<math>\int {1 \over x}\,dx = \ln \left|x \right| + C</math>
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| ::More generally,<ref>"[http://golem.ph.utexas.edu/category/2012/03/reader_survey_logx_c.html Reader Survey: log|''x''| + ''C'']", Tom Leinster, ''The ''n''-category Café'', March 19, 2012</ref>
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| ::<math>\int {1 \over x}\,dx = \begin{cases}\ln \left|x \right| + C^- & x < 0\\
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| \ln \left|x \right| + C^+ & x > 0
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| \end{cases}</math>
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| :<math>\int\frac{c}{ax + b} \, dx= \frac{c}{a}\ln\left|ax + b\right| + C</math>
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| ===Exponential functions===
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| :''More integrals: [[List of integrals of exponential functions]]''
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| :<math>\int e^x\,dx = e^x + C</math>
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| :<math>\int f'(x)e^{f(x)}\,dx = e^{f(x)} + C</math>
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| :<math>\int a^x\,dx = \frac{a^x}{\ln a} + C</math>
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| ===Logarithms===
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| :''More integrals: [[List of integrals of logarithmic functions]]''
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| :<math>\int \ln x\,dx = x \ln x - x + C</math>
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| :<math>\int \log_a x\,dx = x\log_a x - \frac{x}{\ln a} + C</math>
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| ===Trigonometric functions===
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| :''More integrals: [[List of integrals of trigonometric functions]]''
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| :<math>\int \sin{x}\, dx = -\cos{x} + C</math>
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| :<math>\int \cos{x}\, dx = \sin{x} + C</math>
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| :<math>\int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C = \ln{\left| \sec{x} \right|} + C</math>
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| :<math>\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C</math>
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| :<math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C</math>
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| :: (See [[Integral of the secant function]]. This result was a well-known conjecture in the 17th century.)
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| :<math>\int \csc{x} \, dx = \ln{\left| \csc{x} - \cot{x}\right|} + C</math>
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| :<math>\int \sec^2 x \, dx = \tan x + C</math>
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| :<math>\int \csc^2 x \, dx = -\cot x + C</math>
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| :<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C</math>
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| :<math>\int \csc{x} \, \cot{x} \, dx = -\csc{x} + C</math>
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| :<math>\int \sin^2 x \, dx = \frac{1}{2}\left(x - \frac{\sin 2x}{2} \right) + C = \frac{1}{2}(x - \sin x\cos x ) + C </math>
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| :<math>\int \cos^2 x \, dx = \frac{1}{2}\left(x + \frac{\sin 2x}{2} \right) + C = \frac{1}{2}(x + \sin x\cos x ) + C </math>
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| :<math>\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C</math>
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| :: (see [[integral of secant cubed]])
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| :<math>\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx</math>
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| :<math>\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx</math>
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| ===Inverse trigonometric functions===
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| :''More integrals: [[List of integrals of inverse trigonometric functions]]''
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| :<math>\int \arcsin{x} \, dx = x \arcsin{x} + \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le +1 </math>
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| :<math>\int \arccos{x} \, dx = x \arccos{x} - \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le +1 </math>
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| :<math>\int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x </math>
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| :<math>\int \arccot{x} \, dx = x \arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x </math>
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| :<math>\int \arcsec{x} \, dx = x \arcsec{x} - \ln \vert x \, ( 1 + \sqrt{ 1 - x^{-2} } \, ) \vert + C , \text{ for } \vert x \vert \ge +1 </math>
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| :<math>\int \arccsc{x} \, dx = x \arccsc{x} + \ln \vert x \, ( 1 + \sqrt{ 1 - x^{-2} } \, ) \vert + C , \text{ for } \vert x \vert \ge +1 </math>
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| ===Hyperbolic functions===
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| :''More integrals: [[List of integrals of hyperbolic functions]]''
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| :<math>\int \sinh x \, dx = \cosh x + C</math>
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| :<math>\int \cosh x \, dx = \sinh x + C</math>
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| :<math>\int \tanh x \, dx = \ln \cosh x + C</math>
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| :<math>\int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 </math>
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| :<math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math>
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| :<math>\int \operatorname{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C , \text{ for } x \neq 0 </math>
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| ===Inverse hyperbolic functions===
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| :''More integrals: [[List of integrals of inverse hyperbolic functions]]''
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| :<math>\int \operatorname{arsinh} \, x \, dx = x \, \operatorname{arsinh} \, x - \sqrt{ x^2 + 1 } + C , \text{ for all real } x </math>
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| :<math>\int \operatorname{arcosh} \, x \, dx = x \, \operatorname{arcosh} \, x - \sqrt{ x^2 - 1 } + C , \text{ for } x \ge 1 </math>
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| :<math>\int \operatorname{artanh} \, x \, dx = x \, \operatorname{artanh} \, x + \frac{\ln\left(\,1-x^2\right)}{2} + C , \text{ for } \vert x \vert < 1 </math>
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| :<math>\int \operatorname{arcoth} \, x \, dx = x \, \operatorname{arcoth} \, x + \frac{\ln\left(x^2-1\right)}{2} + C , \text{ for } \vert x \vert > 1 </math>
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| :<math>\int \operatorname{arsech} \, x \, dx = x \, \operatorname{arsech} \, x + \arcsin x + C , \text{ for } 0 < x \le 1 </math>
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| :<math>\int \operatorname{arcsch} \, x \, dx = x \, \operatorname{arcsch} \, x + \vert \operatorname{arsinh} \, x \vert + C , \text{ for } x \neq 0 </math>
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| ===Products of functions proportional to their second derivatives===
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| :<math>\int \cos ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( a\sin ax + b\cos ax \right) + C</math>
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| :<math>\int \sin ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( b\sin ax - a\cos ax \right) + C</math>
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| :<math>\int \cos ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( a\sin ax\, \cosh bx+ b\cos ax\, \sinh bx \right) + C</math>
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| :<math>\int \sin ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( b\sin ax\, \sinh bx- a\cos ax\, \cosh bx \right) + C</math>
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| ===Absolute-value functions===
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| Let ''f'' be a function which has at most one root on each interval on which it is defined, and ''g'' an antiderivative of ''f'' that is zero at each root of f (such an antiderivative exists if and only if the condition on ''f'' is satisfied), then
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| :<math>\int \left| f(x)\right|\,dx = \sgn(f(x))g(x)+C,</math>
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| where sgn(''x'') is the [[sign function]], which takes the values -1, 0, 1 when ''x'' is respectively negative, zero or positive. This gives the following formulas (where ''a''≠0):
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| :<math>\int \left| (ax + b)^n \right|\,dx = \sgn(ax + b) {(ax + b)^{n+1} \over a(n+1)} + C \quad [\,n\text{ is odd, and } n \neq -1\,] \,.</math>
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| :<math>\int \left| \tan{ax} \right|\,dx = \frac{-1}{a}\sgn(\tan{ax}) \ln(\left|\cos{ax}\right|) + C</math>
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| when <math>ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) \,</math> for some integer ''n''.
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| :<math>\int \left| \csc{ax} \right|\,dx = \frac{-1}{a}\sgn(\csc{ax}) \ln(\left| \csc{ax} + \cot{ax} \right|) + C </math>
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| when <math>ax \in \left( n\pi, n\pi + \pi \right) \,</math> for some integer ''n''.
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| :<math>\int \left| \sec{ax} \right|\,dx = \frac{1}{a}\sgn(\sec{ax}) \ln(\left| \sec{ax} + \tan{ax} \right|) + C </math>
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| when <math>ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) \,</math> for some integer ''n''. | |
| :<math>\int \left| \cot{ax} \right|\,dx = \frac{1}{a}\sgn(\cot{ax}) \ln(\left|\sin{ax}\right|) + C </math>
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| when <math>ax \in \left( n\pi, n\pi + \pi \right) \,</math> for some integer ''n''.
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| If the function ''f'' does not has any continuous anti-derivative which takes the value zero at the zeros of ''f'' (this is the case for the sine and the cosine functions), then <math>\sgn(f(x)) \int f(x)dx</math> is an anti-derivative of ''f'' on every [[interval (mathematics)|interval]] on which ''f'' is not zero, but may be discontinuous at the points where ''f''(''x'')=0. For having a continuous anti-derivative, one has thus to add a well chosen [[step function]]. If we also use the fact that the absolute values of sine and cosine are periodic with period {{pi}}, then we get:
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| :<math>\int \left| \sin{ax} \right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} \right\rfloor - {1 \over a} \cos{\left( ax - \left\lfloor \frac{ax}{\pi} \right\rfloor \pi \right)} + C\;</math> {{cn|date=April 2013}}
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| :<math>\int \left|\cos {ax}\right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor + {1 \over a} \sin{\left( ax - \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor \pi \right)} + C\;</math> {{cn|date=April 2013}}
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| ===Special functions===
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| Ci, Si: [[Trigonometric integral]]s, Ei: [[Exponential integral]], li: [[Logarithmic integral function]], erf: [[Error function]]
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| : <math>\int \operatorname{Ci}(x) \, dx = x \operatorname{Ci}(x) - \sin x</math>
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| : <math>\int \operatorname{Si}(x) \, dx = x \operatorname{Si}(x) + \cos x</math>
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| : <math>\int \operatorname{Ei}(x) \, dx = x \operatorname{Ei}(x) - e^x</math>
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| : <math>\int \operatorname{li}(x) \, dx = x \operatorname{li}(x)-\operatorname{Ei}(2 \ln x) </math>
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| : <math>\int \frac{\operatorname{li}(x)}{x}\,dx = \ln x\, \operatorname{li}(x) -x </math>
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| : <math>\int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x \operatorname{erf}(x)</math>
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| ==Definite integrals lacking closed-form antiderivatives==
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| There are some functions whose antiderivatives ''cannot'' be expressed in [[Closed-form expression|closed form]]. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.
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| :<math>\int_0^\infty \sqrt{x}\,e^{-x}\,dx = \frac{1}{2}\sqrt \pi</math> (see also [[Gamma function]])
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| :<math>\int_0^\infty e^{-a x^2}\,dx = \frac{1}{2} \sqrt \frac {\pi} {a} </math> for {{math|''a'' > 0}} (the [[Gaussian integral]])
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| :<math>\int_0^\infty{x^2 e^{-a x^2}\,dx} = \frac{1}{4} \sqrt \frac {\pi} {a^3} </math> for {{math|''a'' > 0}}
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| :<math>\int_0^\infty x^{2n} e^{-a x^2}\,dx
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| = \frac{2n-1}{2a} \int_0^\infty x^{2(n-1)} e^{-a x^2}\,dx
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| = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}
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| = \frac{(2n)!}{n! 2^{2n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}
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| </math> for {{math|''a'' > 0}}, {{math|''n''}} is 1, 2, 3, ... and !! is the [[double factorial]].
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| :<math>\int_0^\infty{x^3 e^{-a x^2}\,dx} = \frac{1}{2 a^2} </math> when {{math|''a'' > 0}}
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| :<math>\int_0^\infty x^{2n+1} e^{-a x^2}\,dx
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| = \frac {n} {a} \int_0^\infty x^{2n-1} e^{-a x^2}\,dx
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| = \frac{n!}{2 a^{n+1}}
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| </math> for {{math|''a'' > 0}}, {{math|''n'' {{=}} 0, 1, 2, ....}}
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| :<math>\int_0^\infty \frac{x}{e^x-1}\,dx = \frac{\pi^2}{6}</math> (see also [[Bernoulli number]])
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| :<math>\int_0^\infty \frac{x^2}{e^x-1}\,dx = 2\zeta(3) \simeq 2.40</math>
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| :<math>\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}</math>
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| :<math>\int_0^\infty \frac{\sin{x}}{x}\,dx = \frac{\pi}{2}</math> (see [[sinc function]] and [[Sine integral]])
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| :<math>\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx = \frac{\pi}{2}</math>
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| :<math>\int_0^\frac{\pi}{2}\sin^n{x}\,dx = \int_0^\frac{\pi}{2} \cos^n{x}\,dx = \frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2}</math> (if {{math|''n''}} is an even integer and {{math|''n'' ≥ 2}})
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| :<math>\int_0^\frac{\pi}{2}\sin^n{x}\,dx = \int_0^\frac{\pi}{2}\cos^n{x}\,dx = \frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n}</math> (if {{math|''n''}} is an odd integer and {{math|''n'' ≥ 3}})
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| :<math>\int_{-\pi}^\pi \cos(\alpha x)\cos^n(\beta x) dx = \begin{cases}
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| \frac{2 \pi}{2^n} \binom{n}{m} & |\alpha|= |\beta (2m-n)| \\
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| 0 & \text{otherwise}
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| \end{cases} </math> (for {{math|''α '', ''β'', ''m'', ''n''}} integers with {{math|''β'' ≠ 0}} and {{math|''m'', ''n'' ≥ 0}}, see also [[Binomial coefficient]])
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| :<math>\int_{-\pi}^\pi \sin(\alpha x) \cos^n(\beta x) dx = 0</math> (for {{math|''α '', ''β''}} real and {{math|''n''}} non-negative integer, see also [[Symmetry]])
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| :<math>\int_{-\pi}^\pi \sin(\alpha x) \sin^n(\beta x) dx = \begin{cases}
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| (-1)^{(n+1)/2} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \text{ odd},\ \alpha = \beta (2m-n) \\
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| 0 & \text{otherwise}
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| \end{cases} </math> (for {{math|''α '', ''β'', ''m'', ''n''}} integers with {{math|''β'' ≠ 0}} and {{math|''m'', ''n'' ≥ 0}}, see also [[Binomial coefficient]])
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| :<math>\int_{-\pi}^{\pi} \cos(\alpha x) \sin^n(\beta x) dx = \begin{cases}
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| (-1)^{n/2} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \text{ even},\ |\alpha| = |\beta (2m-n)| \\
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| 0 & \text{otherwise}
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| \end{cases} </math> (for {{math|''α '', ''β'', ''m'', ''n''}} integers with {{math|''β'' ≠ 0}} and {{math|''m'', ''n'' ≥ 0}}, see also [[Binomial coefficient]])
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| :<math>\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx = \sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]</math> (where {{math|exp[''u'']}} is the [[exponential function]] {{math|e<sup>''u''</sup>}}, and {{math|''a'' > 0}})
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| :<math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (where <math>\Gamma(z)</math> is the [[Gamma function]])
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| :<math>\int_0^1 x^{\alpha-1}(1-x)^{\beta-1} dx = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} </math> (for {{math|Re(''α '') > 0}} and {{math|Re(''β'') > 0}}, see [[Beta function]])
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| :<math>\int_0^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (where {{math|''I''<sub>0</sub>(''x'')}} is the modified [[Bessel function]] of the first kind)
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| :<math>\int_0^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right) </math>
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| :<math>\int_{-\infty}^\infty (1 + x^2/\nu)^{-(\nu + 1)/2}\,dx = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2)}</math> (for {{math|''ν'' > 0}} , this is related to the [[probability density function]] of the [[Student's t-distribution]])
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| If the function ''f'' has [[bounded variation]] on the interval [''a'',''b''], then the [[method of exhaustion]] provides a formula for the integral:
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| :<math>\int_a^b{f(x)\,dx} = (b - a) \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)2^{-n} ).</math>
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| :<math>\int_0^1 \ln(1/x)^p\,dx = p!\;</math>
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| {{hidden
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| |(Click "show" at right to see a proof or "hide" to hide it.)
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| |2=
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| ----
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| The change of variable <math>x = e^{-t}</math> gives
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| : <math>I_p = \int_0^1 \ln(1/x)^p\;dx = \int_0^\infty t^pe^{-t}dt,</math>
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| and, under this form the result appears in [[List of integrals of exponential functions#Definite integrals]]
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| ----
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| }}
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| The "[[sophomore's dream]]"
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| :<math>\begin{align}
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| \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n} &&(= 1.29128599706266\dots)\\
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| \int_0^1 x^x \,dx &= -\sum_{n=1}^\infty (-n)^{-n} &&(= 0.78343051071213\dots)
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| \end{align}</math>
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| attributed to [[Johann Bernoulli]].
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| == See also ==
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| * [[Indefinite sum]]
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| * [[Incomplete Gamma function]]
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| * [[List of mathematical series]]
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| * [[List of limits]]
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| * [[Symbolic integration]]
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| == References ==
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| {{reflist}}
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| * [[Milton Abramowitz|M. Abramowitz]] and [[Irene Stegun|I.A. Stegun]], editors. ''[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]''.
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| * I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. ''Table of Integrals, Series, and Products'', seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. [http://www.mathtable.com/gr Errata.] ''(Several previous editions as well.)''
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| * A.P. Prudnikov (А.П. Прудников), Yu.A. Brychkov (Ю.А. Брычков), O.I. Marichev (О.И. Маричев). ''Integrals and Series''. First edition (Russian), volume 1–5, [[Nauka (publisher)|Nauka]], 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/[[CRC Press]], 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
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| * Yu.A. Brychkov (Ю.А. Брычков), ''Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas''. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
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| * Daniel Zwillinger. ''CRC Standard Mathematical Tables and Formulae'', 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. ''(Many earlier editions as well.)''
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| === Historical ===
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| * Meyer Hirsch, [http://books.google.com/books?id=Cdg2AAAAMAAJ Integraltafeln, oder, Sammlung von Integralformeln] (Duncker und Humblot, Berlin, 1810)
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| * Meyer Hirsch, [http://books.google.com/books?id=NsI2AAAAMAAJ Integral Tables, Or, A Collection of Integral Formulae] (Baynes and son, London, 1823) [English translation of ''Integraltafeln'']
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| * David Bierens de Haan, [http://www.archive.org/details/nouvetaintegral00haanrich Nouvelles Tables d'Intégrales définies] (Engels, Leiden, 1862)
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| * Benjamin O. Pierce [http://books.google.com/books?id=pYMRAAAAYAAJ A short table of integrals - revised edition] (Ginn & co., Boston, 1899)
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| == External links ==
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| === Tables of integrals ===
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| * [http://tutorial.math.lamar.edu/pdf/Common_Derivatives_Integrals.pdf Paul's Online Math Notes]
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| * A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): [http://pi.physik.uni-bonn.de/~dieckman/IntegralsIndefinite/IndefInt.html Indefinite Integrals] [http://pi.physik.uni-bonn.de/~dieckman/IntegralsDefinite/DefInt.html Definite Integrals]
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| * [http://mathmajor.org/calculus-and-analysis/table-of-integrals/ Math Major: A Table of Integrals]
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| * {{cite web | last1=O'Brien |first1=Francis J. Jr. | url=http://www.docstoc.com/docs/23969109/500-Integrals-of-Elementary-and-Special-Functions |title=500 Integrals}} Derived integrals of exponential and logarithmic functions
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| * [http://www.apmaths.uwo.ca/RuleBasedMathematics/index.html Rule-based Mathematics] Precisely defined indefinite integration rules covering a wide class of integrands
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| * {{cite arxiv| first1= Richard J. | last1=Mathar | title=Yet another table of integrals | eprint=1207.5845 |year=2012}}
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| === Derivations ===
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| * [http://www.math.tulane.edu/~vhm/Table.html V. H. Moll, The Integrals in Gradshteyn and Ryzhik]
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| === Online service ===
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| * [http://www.wolframalpha.com/examples/Integrals.html Integration examples for Wolfram Alpha]
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| === Open source programs ===
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| *[http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page wxmaxima gui for Symbolic and numeric resolution of many mathematical problems]
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| [[Category:Integrals|*]]
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| [[Category:Mathematics-related lists|Integrals]]
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| [[Category:Mathematical tables|Integrals]]
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