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Topics in Calculus

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Integration is one of the two basic operations in calculus. While differentiation has easy rules by which the derivative of a complicated 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.

Historical development of integrals

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.

Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, 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.

Lists of integrals

More detail may be found on the following pages for the lists of integrals:

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 Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms). 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.

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.

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 Wolfram Mathematica Online Integrator.

Integrals of simple functions

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.

These formulas only state in another form the assertions in the table of derivatives.

Rational functions

more integrals: List of integrals of rational functions

$\int a\,dx = ax + C$

$\int x^a\,dx = \frac{x^{a+1}}{a+1} + C$

$\int {1 \over x}\,dx = \ln \left|x \right| + C$

Exponential functions

more integrals: List of integrals of exponential functions

$\int e^x\,dx = e^x + C$

$\int a^x\,dx = \frac{a^x}{\ln a} + C$

Logarithms

more integrals: List of integrals of logarithmic functions

$\int \ln x\,dx = x \ln x - x + C$

$\int \log_a x\,dx = x\log_a x - \frac{x}{\ln a} + C$

Trigonometric functions

more integrals: List of integrals of trigonometric functions

$\int \sin{x}\, dx = -\cos{x} + C$

$\int \cos{x}\, dx = \sin{x} + C$

$\int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C = \ln{\left| \sec{x} \right|} + C$

$\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C$

$\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C$

$\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C$

$\int \sec^2 x \, dx = \tan x + C$

$\int \csc^2 x \, dx = -\cot x + C$

$\int \sec{x} \, \tan{x} \, dx = \sec{x} + C$

$\int \csc{x} \, \cot{x} \, dx = -\csc{x} + C$

$\int \sin^2 x \, dx = \frac{1}{2}\left(x - \frac{\sin 2x}{2} \right) + C = \frac{1}{2}(x - \sin x\cos x ) + C$

$\int \cos^2 x \, dx = \frac{1}{2}\left(x + \frac{\sin 2x}{2} \right) + C = \frac{1}{2}(x + \sin x\cos x ) + C$

$\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C$

(see integral of secant cubed)

$\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx$

$\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx$

Inverse trigonometric functions

more integrals: List of integrals of inverse trigonometric functions

$\int \arcsin{x} \, dx = x \, \arcsin{x} + \sqrt{1 - x^2} + C$

$\int \arccos{x} \, dx = x \, \arccos{x} - \sqrt{1 - x^2} + C$

$\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C$

$\int \arcsec{x} \, dx = x \, \arcsec{x} - \frac{\sqrt{x^2 - 1}\ln{(x + \sqrt{x^2-1})}}{x \, \sqrt{1 - \frac{1}{x^2}}} + C$

$\int \arccsc{x} \, dx = x \, \arccsc{x} + \frac{\sqrt{x^2 - 1}\ln{(x + \sqrt{x^2-1})}}{x \, \sqrt{1 - \frac{1}{x^2}}} + C$

$\int \arccot{x} \, dx = x \, \arccot{x} + \frac{1}{2} \ln{\left| 1 + x^2\right|} + C$

Hyperbolic functions

more integrals: List of integrals of hyperbolic functions

$\int \sinh x \, dx = \cosh x + C$

$\int \cosh x \, dx = \sinh x + C$

$\int \tanh x \, dx = \ln| \cosh x | + C$

$\int \mbox{cosech}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C$

$\int \mbox{sech}\,x \, dx = \arctan\,(\sinh x) + C$

$\int \coth x \, dx = \ln| \sinh x | + C$

$\int \mbox{sech}^2 x\, dx = \tanh x + C$

Inverse hyperbolic functions

more integrals: List of integrals of inverse hyperbolic functions

$\int \operatorname{arsinh}\, x \, dx = x\, \operatorname{arsinh}\, x - \sqrt{x^2+1} + C$

$\int \operatorname{arcosh}\, x \, dx = x\, \operatorname{arcosh}\, x - \sqrt{x^2-1} + C$

$\int \operatorname{artanh}\, x \, dx = x\, \operatorname{artanh}\, x + \frac{1}{2}\ln{(1-x^2)} + C$

$\int \operatorname{arcsch}\,x \, dx = x\, \operatorname{arcsch}\, x+ \ln{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C$

$\int \operatorname{arsech}\,x \, dx = x\, \operatorname{arsech}\, x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C$

$\int \operatorname{arcoth}\,x \, dx = x\, \operatorname{arcoth}\, x+ \frac{1}{2}\ln{(x^2-1)} + C$

Composed functions

$\int \cos ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( a\sin ax + b\cos ax \right) + C$

$\int \sin ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( b\sin ax - a\cos ax \right) + C$

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

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

Absolute value functions

$\int \left| (ax + b)^n \right|\,dx = {(ax + b)^{n+2} \over a(n+1) \left| ax + b \right|} + C \,\, [\,n\text{ is odd, and } n \neq -1\,]$

$\int \left| \sin{ax} \right|\,dx = {-1 \over a} \left| \sin{ax} \right| \cot{ax} + C$

$\int \left| \cos{ax} \right|\,dx = {1 \over a} \left| \cos{ax} \right| \tan{ax} + C$

$\int \left| \tan{ax} \right|\,dx = {\tan(ax)[-\ln\left|\cos{ax}\right|] \over a \left| \tan{ax} \right|} + C$

$\int \left| \csc{ax} \right|\,dx = {-\ln \left| \csc{ax} + \cot{ax} \right|\sin{ax} \over a \left| \sin{ax} \right|} + C$

$\int \left| \sec{ax} \right|\,dx = {\ln \left| \sec{ax} + \tan{ax} \right| \cos{ax} \over a \left| \cos{ax} \right|} + C$

$\int \left| \cot{ax} \right|\,dx = {\tan(ax)[\ln\left|\sin{ax}\right|] \over a \left| \tan{ax} \right|} + C$

Special functions

$\int \operatorname{Ci}(x) dx = x\,\operatorname{Ci}(x) - \sin x$

$\int \operatorname{Si}(x) dx = x\,\operatorname{Si}(x) + \cos x$

$\int \operatorname{Ei}(x) dx = x\,\operatorname{Ei}(x) - e^x$

$\int \operatorname{li}(x)dx = x\, \operatorname{li}(x)-\operatorname{Ei}(2 \ln x)$

$\int \frac{\operatorname{li}(x)}{x}\,dx = \ln x\, \operatorname{li}(x) -x$

$\int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x\, \text{erf}(x)$

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.

$\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi$ (see also Gamma function)

$\int_0^\infty{e^{-a x^2}\,dx} = \frac{1}{2} \sqrt \frac {\pi} {a}$ (the Gaussian integral)

$\int_0^\infty{x^2 e^{-a x^2}\,dx} = \frac{1}{4} \sqrt \frac {\pi} {a^3}$ when a > 0

$\int_0^\infty{x^{2n} e^{-a x^2}\,dx} = \frac{2n-1}{2a} \int_0^\infty{x^{2(n-1)} e^{-a x^2}\,dx} = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}$ when a > 0, n is 1,2,3,... and !! is the double factorial

$\int_0^\infty{x^3 e^{-a x^2}\,dx} = \frac{1}{2 a^2}$ when a > 0

$\int_0^\infty{x^{2n+1} e^{-a x^2}\,dx} = \frac {n} {a} \int_0^\infty{x^{2n-1} e^{-a x^2}\,dx} = \frac{n!}{2 a^{n+1}}$ when a > 0, n is 1, 2, ....

$\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}$ (see also Bernoulli number)

$\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}$

$\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}$

$\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2}$ (if n is an even integer and $\scriptstyle{n \ge 2}$ )

$\int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n}$ (if $\scriptstyle{n}$ is an odd integer and $\scriptstyle{n \ge 3}$ )

$\int_{-\pi}^{\pi} \cos(\alpha x)\cos^n(\beta x) dx = \left \{ \begin{array}{cc} \frac{2 \pi}{2^n} \binom{n}{m} & |\alpha|= |\beta (2m-n)| \\ 0 & \mbox{otherwise} \\ \end{array} \right .$ (for $\scriptstyle \alpha, \beta, m, n$ integers with $\scriptstyle \beta \neq 0$ and $\scriptstyle m, n \geq 0$ , see also Binomial coefficient)

$\int_{-\pi}^{\pi} \sin(\alpha x) \cos^n(\beta x) dx = 0$ (for $\scriptstyle \alpha,\beta$ real and $\scriptstyle n$ non-negative integer, see also Symmetry)

$\int_{-\pi}^{\pi} \sin(\alpha x) \sin^n(\beta x) dx = \left \{ \begin{array}{cc} (-1)^{(n+1)/2} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \mbox{ odd},\ \alpha = \beta (2m-n) \\ 0 & \mbox{otherwise} \\ \end{array} \right .$ (for $\scriptstyle \alpha, \beta, m, n$ integers with $\scriptstyle \beta \neq 0$ and $\scriptstyle m, n \geq 0$ , see also Binomial coefficient)

$\int_{-\pi}^{\pi} \cos(\alpha x) \sin^n(\beta x) dx = \left \{ \begin{array}{cc} (-1)^{n/2} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \mbox{ even},\ |\alpha| = |\beta (2m-n)| \\ 0 & \mbox{otherwise} \\ \end{array} \right .$ (for $\scriptstyle \alpha, \beta, m, n$ integers with $\scriptstyle \beta \neq 0$ and $\scriptstyle m,n \geq 0$ , see also Binomial coefficient)

$\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}$

$\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)$ (where

Γ(z)

is the Gamma function)

$\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]$ (where

exp[u]

is the exponential function

e u

, and

a > 0

)

$\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)$ (where

I 0 (x)

is the modified Bessel function of the first kind)

$\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right)$

$\int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\,$ , $\nu > 0\,$ , this is related to the probability density function of the Student's t-distribution)

The method of exhaustion provides a formula for the general case when no antiderivative exists:

$\int_a^b{f(x)\,dx} = (b - a) \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)2^{-n} ).$

$\int_0^1 [\ln(1/x)]^p\,dx = p!$

The "sophomore's dream"

$\begin{align} \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n} &&(= 1.29128599706266\dots)\\ \int_0^1 x^x \,dx &= -\sum_{n=1}^\infty (-1)^nn^{-n} &&(= 0.783430510712\dots) \end{align}$

attributed to Johann Bernoulli.

References

M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.

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. Errata. (Several previous editions as well.)

A.P. Prudnikov (А.П. Прудников), Yu.A. Brychkov (Ю.А. Брычков), O.I. Marichev (О.И. Маричев). Intelgrals and Series. First edition (Russian), volume 1–5, 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.

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.)

Historical

Meyer Hirsch, Integraltafeln, oder, Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810)
Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
David de Bierens de Haan, Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862)
Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)

External links

Tables of integrals

S.O.S. Mathematics: Tables and Formulas (warning: may serve popunders)
Paul's Online Math Notes
A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
O'Brien, Francis J. Jr. 500 Integrals Derived integrals of exponential and logarithmic functions

Derivations

V. H. Moll, The Integrals in Gradshteyn and Ryzhik

Online service

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Lists of integrals

Contents

Historical development of integrals

Lists of integrals

Integrals of simple functions

Rational functions

Exponential functions

Logarithms

Trigonometric functions

Inverse trigonometric functions

Hyperbolic functions

Inverse hyperbolic functions

Composed functions

Absolute value functions

Special functions

Definite integrals lacking closed-form antiderivatives

The "sophomore's dream"

References

Historical

See also

External links

Tables of integrals

Derivations

Online service

	List of integrals of inverse hyperbolic functions
	List of integrals of logarithmic functions
	List of integrals of hyperbolic functions

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