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#REDIRECT [[Lists of integrals]]
[[Integral|Integration]] is one of the two basic operations in [[calculus]] and since it, unlike [[derivative|differentiation]], is non-trivial, tables of known integrals are often useful.
This page lists some of the most common antiderivatives; a more complete list can be found in the [[list of integrals]].
We use ''C'' 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.
These formulas only state in another form the assertions in the [[table of derivatives]].
==Rules for integration of general functions==
:<math>\int af(x)\,dx = a\int f(x)\,dx</math>
:<math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx</math>
:<math>\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int [f'(x) \left(\int g(x)\,dx\right)]\,dx</math>
==Integrals of simple functions==
===[[Rational function]]s===
:''more integrals: [[List of integrals of rational functions]]''
:<math>\int \,dx = x + C</math>
:<math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ if }n \ne -1</math>
:<math>\int \frac{1}{x}\,dx = \ln{\left|x\right|} + C</math>
:<math>\int {dx \over {a^2+(bx)^2}} = {1 \over ab}\arctan {bx \over a} + C</math>
===[[Irrational function]]s===
:''more integrals: [[List of integrals of irrational functions]]''
:<math>\int {dx \over \sqrt{a^2-x^2}} = \arcsin {x \over a} + C</math>
:<math>\int {-dx \over \sqrt{a^2-x^2}} = \arccos {x \over a} + C</math>
:<math>\int {dx \over x\sqrt{x^2-a^2}} = {1 \over a}\mbox{arcsec}\,{|x| \over a} + C</math>
===[[Logarithm]]s===
:''more integrals: [[List of integrals of logarithmic functions]]''
:<math>\int \ln {x}\,dx = x \ln {x} - x + C</math>
:<math>\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C</math>
===[[Exponential function]]s===
:''more integrals: [[List of integrals of exponential functions]]''
:<math>\int e^x\,dx = e^x + C</math>
:<math>\int a^x\,dx = \frac{a^x}{\ln{a}} + C</math>
===[[Trigonometric function]]s===
:''more integrals: [[List of integrals of trigonometric functions]] and [[List of integrals of arc functions]]''
:<math>\int \sin{x}\, dx = -\cos{x} + C</math>
:<math>\int \cos{x}\, dx = \sin{x} + C</math>
:<math>\int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C</math>
:<math>\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C</math>
:<math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C</math>
:<math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C</math>
:<math>\int \sec^2 x \, dx = \tan x + C</math>
:<math>\int \csc^2 x \, dx = -\cot x + C</math>
:<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C</math>
:<math>\int \csc{x} \, \cot{x} \, dx = - \csc{x} + C</math>
:<math>\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C</math>
:<math>\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C</math>
:<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>
:<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>
:<math>\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C</math>
===[[Hyperbolic function]]s===
:''more integrals: [[List of integrals of hyperbolic functions]]''
:<math>\int \sinh x \, dx = \cosh x + C</math>
:<math>\int \cosh x \, dx = \sinh x + C</math>
:<math>\int \tanh x \, dx = \ln |\cosh x| + C</math>
:<math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C</math>
:<math>\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C</math>
:<math>\int \coth x \, dx = \ln|\sinh x| + C</math>
===[[Inverse hyperbolic function]]s===
: <math>\int \sinh^{-1} x \, dx = x \sinh^{-1} x+ \sqrt{x^2+1} + C</math>
: <math>\int \cosh^{-1} x \, dx = x \cosh^{-1} x+ \sqrt{x^2-1} + C</math>
: <math>\int \tanh^{-1} x \, dx = x \tanh^{-1} x+ \frac{1}{2}\log{(1-x^2)} + C</math>
:<math>\int \mbox{csch}^{-1}\,x \, dx = x \mbox{csch}^{-1}\ x+ \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C</math>
:<math>\int \mbox{sech}^{-1}\,x \, dx = x \mbox{sech}^{-1}\ x- \tan^{-1}{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C</math>
: <math>\int \coth^{-1} x \, dx = x \coth^{-1} x+ \frac{1}{2}\log{(x^2-1)} + C</math>
==Definite integrals lacking closed-form antiderivatives==
There are some functions whose antiderivatives ''cannot'' be expressed in 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.
:<math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi</math> (see also [[Gamma function]])
:<math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi</math> (the [[Gaussian integral]])
:<math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}</math> (see also [[Bernoulli number]])
:<math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}</math>
:<math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}</math>
:<math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (where <math>\Gamma(z)</math> is the [[Gamma function]])
:<math>\int_{-\infty}^\infty \exp\left[-(ax^2+bx+c)\right]\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]</math>(where <math> \exp\left[f(x)\right] = e^{f(x)}</math>.)
:<math>\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (where <math>I_{0}(x)</math> is the modified [[Bessel function]] of the first kind)
:<math>\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \sqrt{x^2 + y^2} </math>
The [[method of exhaustion]] provides a formula for the general case when no antiderivative exists:
:<math>\int\limits_a^b {f(x)dx = \left( {b - a} \right)} \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>.
[[Category:Integrals|*]]
[[Category:Mathematics-related lists|Integrals]]
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