Lists of integrals

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Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions

See the following pages for lists of integrals:

1 Table of Integrals

Table of Integrals

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; 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

These rules apply only whenever the respective functions are integrable.

Failed to parse (Missing texvc executable; please see math/README to configure.): \int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \neq 0 \mbox{, constant)}\,\!

Failed to parse (Missing texvc executable; please see math/README to configure.): \int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx

Failed to parse (Missing texvc executable; please see math/README to configure.): \int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int [f(x)]^n f'(x)\,dx = {[f(x)]^{n+1} \over n+1} + C \qquad\mbox{(for } n\neq -1\mbox{)}\,\!

Integrals of simple functions

Rational functions

more integrals: List of integrals of rational functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \,dx = x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ if }n \ne -1

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {dx \over x} = \ln{\left|x\right|} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {dx \over {a^2+x^2}} = {1 \over a}\arctan {x \over a} + C

Irrational functions

more integrals: List of integrals of irrational functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int {-dx \over x \sqrt{x^2-a^2}} = {1 \over a} \csc^{-1} {|x| \over a} + C

Logarithms

more integrals: List of integrals of logarithmic functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \ln {x}\,dx = x \ln {x} - x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C

Exponential functions

more integrals: List of integrals of exponential functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int e^x\,dx = e^x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int a^x\,dx = \frac{a^x}{\ln{a}} + C

Trigonometric functions

more integrals: List of integrals of trigonometric functions and List of integrals of inverse trigonometric functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sin{x}\, dx = -\cos{x} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \cos{x}\, dx = \sin{x} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \csc{x} \, dx = \ln{\left| \csc{x} - \cot{x}\right|} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sec^2 x \, dx = \tan x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \csc^2 x \, dx = -\cot x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sec{x} \, \tan{x} \, dx = \sec{x} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \csc{x} \, \cot{x} \, dx = - \csc{x} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \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)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C

Hyperbolic functions

more integrals: List of integrals of hyperbolic functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \sinh x \, dx = -\cosh x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \cosh x \, dx = \sinh x + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \tanh x \, dx = \ln| \cosh x | + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \coth x \, dx = \ln| \sinh x | + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \mbox{sech}^2 x\, dx = \tanh x + C

Inverse hyperbolic functions

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \operatorname{arcsinh} x \, dx = x \operatorname{arcsinh} x - \sqrt{x^2+1} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \operatorname{arccosh} x \, dx = x \operatorname{arccosh} x - \sqrt{x^2-1} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \operatorname{arctanh} x \, dx = x \operatorname{arctanh} x + \frac{1}{2}\log{(1-x^2)} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \operatorname{arccsch}\,x \, dx = x \operatorname{arccsch} x+ \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \operatorname{arcsech}\,x \, dx = x \operatorname{arcsech} x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \operatorname{arccoth}\,x \, dx = x \operatorname{arccoth} x+ \frac{1}{2}\log{(x^2-1)} + C

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.

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi

 (see also Gamma function)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi

 (the Gaussian integral)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}

 (see also Bernoulli number)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.):   \scriptstyle{n \ge 2}

)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.):  \scriptstyle{n} 
is an odd integer and Failed to parse (Missing texvc executable; please see math/README to configure.):   \scriptstyle{n \ge 3}

)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)

 (where Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma(z)
is the Gamma function)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right]

 (where Failed to parse (Missing texvc executable; please see math/README to configure.): \exp[u]
is the exponential function Failed to parse (Missing texvc executable; please see math/README to configure.): e^u

, and Failed to parse (Missing texvc executable; please see math/README to configure.): a>0 )

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)

 (where Failed to parse (Missing texvc executable; please see math/README to configure.): I_{0}(x)
is the modified Bessel function of the first kind)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right)

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\,

, Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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} )

The "sophomore's dream"

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n} &&(= 1.291285997\dots)\\ \int_0^1 x^x \,dx &= \sum_{n=1}^\infty -(-1)^nn^{-n} &&(= 0.783430510712\dots) \end{align}

(attributed to Johann Bernoulli; see sophomore's dream).

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 Rhyzik. In Gradshteyn and Rhyzik, integrals originating from the book by de Bierens are denoted by BI. Since 1968 there is the Risch algorithm for determining indefinite integrals.

Other lists of integrals

Gradshteyn and Ryzhik contains a large collection of results. Other useful resources include the CRC Standard Mathematical Tables and Formulae and Abramowitz and Stegun. A&S contains many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. There are several web sites which have tables of integrals and integrals on demand.

References

Milton Abramowitz and Irene A. Stegun, eds. 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.)

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.)

External links

Integrals on demand

The Integrator (at Wolfram Research)
TILU Table of Integrals Look Up (maintained by Richard Fateman)

Tables of integrals

S.O.S. Mathematics: Tables and Formulas

Historical

Meyer Hirsch, Integraltafeln, oder, Sammlung von Integralformeln (Duncker un 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)bs:Spisak integrala

es:Anexo:Lista de integrales gl:Lista de integrais hr:Popisi integrala lt:Integralų lentelė pl:Tablica całek ru:Список интегралов

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