Lists of integrals explained

Integration is the basic operation in integral calculus. While differentiation has straightforward 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 (also spelled Meyer Hirsch) in 1810.^[1] These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his Tables d'intégrales définies, supplemented by Supplément aux tables d'intégrales définies in ca. 1864. A new edition was published in 1867 under the title Nouvelles tables d'intégrales définies.

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 Bierens de Haan are denoted by BI.

Not all closed-form expressions 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 which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is, whose antiderivative is (up to constants) the error function.

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:

• List of integrals of rational functions
• List of integrals of irrational functions
• List of integrals of trigonometric functions
• List of integrals of inverse trigonometric functions
• List of integrals of hyperbolic functions
• List of integrals of inverse hyperbolic functions
• List of integrals of exponential functions
• List of integrals of logarithmic functions
• List of integrals of Gaussian functions

Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll's (GR) 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 or Bronshtein and Semendyayev's Guide Book to Mathematics, Handbook of Mathematics or Users' Guide to Mathematics, 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 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.

Integrals with a singularity

When there is a singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), 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 in general, not necessary. For instance in$$\backslash int\; \backslash ,dx\; =\; \backslash ln\; \backslash left|x\; \backslash right|\; +\; C$$there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −i when using a path above the origin and i 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:^[2]

Rational functions

See also: List of integrals of rational functions.

\int adx = ax + C

The following function has a non-integrable singularity at 0 for :

\int x ^{ndx} = (x ^{n+1})/(n +1) + C (for n ≠ -1 ) (Cavalieri's quadrature formula)\int (ax + b) ^{n} dx = ((ax + b) ^{n+1})/(a(n +1 )) + C (for n ≠ -1 ) \int { 1 \over x}dx = ln \left|x \right| + C

• • More generally,^[3] $$\backslash int\; \backslash ,dx\; =\; \backslash begin$$

\ln \left|x \right| + C^- & x < 0\\\ln \left|x \right| + C^+ & x > 0\end

\int (c)/(ax + b) dx = (c)/(a) ln \left|ax + b\right| + C

Exponential functions

See also: List of integrals of exponential functions.

\int e ^{ax} dx = (1)/(a) e ^{ax} + C \int f'(x)e ^{f(x)} dx = e ^{f(x)} + C \int a ^{xdx} = (a ^{x})/(ln a) + C \int{e ^{x} \left( f\left( x \right) + f'\left( x \right) \right)dx} = e ^{x} f\left( x \right) + C \int {e ^{x} \left( f\left( x \right) - \left( - 1 \right) ^{n} (d ^{n} f\left( x \right))/(dx ^{n}) \right)dx} = e ^{x} \sum ^{n} _{k =1} {\left( - 1 \right) ^{k} (d ^{k} f\left( x \right))/(dx ^{k}) } + C(if n is a positive integer)\int {e ^{-} \left( f\left( x \right) - (d ^{n} f\left( x \right))/(dx ^{n}) \right) dx} = - e ^{-} \sum ^{n} _{k =1} (d ^{k} f\left( x \right))/(dx ^{k}) + C (if n is a positive integer)

Logarithms

See also: List of integrals of logarithmic functions.

\int ln xdx = x ln x - x + C = x ( ln x - 1 ) + C \int log _{a} xdx = x log _{a} x - (x)/(ln a) + C = (x)/(ln a) ( ln x - 1 ) + C

Trigonometric functions

See also: 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| \sec{x} \right|} + C = -ln {\left| \cos {x} \right|} + C \int \cot{x} dx = -ln {\left| \csc{x} \right|} + C = ln {\left| \sin{x} \right|} + C \int \sec{x} dx = ln {\left| \sec{x} + \tan{x}\right|} + C = ln \left| \tan\left(\dfrac{x}{ 2 } + \dfrac{\pi}{ 4 }\right) \right| + C

• • (See Integral of the secant function. This result was a well-known conjecture in the 17th century.)

\int \csc{x} dx = -ln {\left| \csc{x} + \cot{x}\right|} + C = ln {\left| \csc{x} - \cot{x}\right|} + C = ln {\left| \tan { (x)/(2) } \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 = (1)/(2) \left(x - (\sin 2 x)/(2) \right) + C = (1)/(2) (x - \sin x\cos x ) + C \int \cos ^{2} x dx = (1)/(2) \left(x + (\sin 2 x)/(2) \right) + C = (1)/(2) (x + \sin x\cos x ) + C \int \tan ^{2} x dx = \tan x - x + C \int \cot ^{2} x dx = - \cot x - x + C \int \sec ^{3} x dx = (1)/(2) (\sec x \tan x + ln |\sec x + \tan x|) + C

• • (See integral of secant cubed.)

\int \csc ^{3} x dx = (1)/(2) ( - \csc x \cot x + ln |\csc x - \cot x|) + C = (1)/(\left( ln \left|\tan 2) (x)/(2) \right| - \csc x \cot x \right) + C \int \sin ^{n} x dx = - (\sin ^{n-1} {x)/(\cos) {x}}{n} + (n -1)/(n) \int \sin ^{n-2} {x} dx \int \cos ^{n} x dx = (\cos ^{n-1} {x)/(\sin) {x}}{n} + (n -1)/(n) \int \cos ^{n-2} {x} dx

Inverse trigonometric functions

See also: List of integrals of inverse trigonometric functions.

\int \arcsin{x} dx = x \arcsin{x} + \sqrt{ 1 - x ^{2}} + C , for \vert x \vert \le 1 \int \arccos{x} dx = x \arccos{x} - \sqrt{ 1 - x ^{2}} + C , for \vert x \vert \le 1 \int \arctan{x} dx = x \arctan{x} - (1)/(2) ln { \vert 1 + x ^{2} \vert } + C , forallreal x \int \arccot{x} dx = x \arccot{x} + (1)/(2) ln { \vert 1 + x ^{2} \vert } + C , forallreal x \int \arcsec{x} dx = x \arcsec{x} - ln \left\vert x \left( 1 + \sqrt{ 1 - x ^{-2} } \right) \right\vert + C , for \vert x \vert \ge 1 \int \arccsc{x} dx = x \arccsc{x} + ln \left\vert x \left( 1 + \sqrt{ 1 - x ^{-2} } \right) \right\vert + C , for \vert x \vert \ge 1

Hyperbolic functions

See also: 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 \coth x dx = ln | \sinh x | + C , for x ≠ 0 \int \operatorname{sech}x dx = \arctan(\sinh x) + C \int \operatorname{csch}x dx = ln |\operatorname{coth} x - \operatorname{csch} x| + C = ln \left| \tanh {x \over 2 }\right| + C , for x ≠ 0 \int \operatorname{sech} ^{2} x dx = \tanh x + C \int \operatorname{csch} ^{2} x dx = - \operatorname{coth}x + C \int \operatorname{sech}{x} \operatorname{tanh}{x} dx = - \operatorname{sech}{x} + C \int \operatorname{csch}{x} \operatorname{coth}{x} dx = - \operatorname{csch}{x} + C

Inverse hyperbolic functions

See also: List of integrals of inverse hyperbolic functions.

\int \operatorname{arcsinh} x dx = x \operatorname{arcsinh} x - \sqrt{ x ^{2} + 1 } + C , forallreal x \int \operatorname{arccosh} x dx = x \operatorname{arccosh} x - \sqrt{ x ^{2} - 1 } + C , for x \ge 1 \int \operatorname{arctanh} x dx = x \operatorname{arctanh} x + (ln \left( 1- x ^{2\right)})/(2) + C , for \vert x \vert < 1 \int \operatorname{arccoth} x dx = x \operatorname{arccoth} x + (ln \left(x ^{2-1\right)})/(2) + C , for \vert x \vert > 1 \int \operatorname{arcsech} x dx = x \operatorname{arcsech} x + \arcsin x + C , for 0 < x \le 1 \int \operatorname{arccsch} x dx = x \operatorname{arccsch} x + \vert \operatorname{arcsinh} x \vert + C , for x ≠ 0

Products of functions proportional to their second derivatives

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

Absolute-value functions

Let be a continuous function, that has at most one zero. If has a zero, let be the unique antiderivative of that is zero at the root of ; otherwise, let be any antiderivative of . Then$$\backslash int\; \backslash left|\; f(x)\backslash right|\backslash ,dx\; =\; \backslash sgn(f(x))g(x)+C,$$where is the sign function, which takes the values −1, 0, 1 when is respectively negative, zero or positive.

This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on is here for insuring the continuity of the integral.

This gives the following formulas (where), which are valid over any interval where is continuous (over larger intervals, the constant must be replaced by a piecewise constant function):

\int \left| (ax + b) ^{n} \right|dx = sgn (ax + b) {(ax + b) ^{n+1} \over a(n +1 )} + C when is odd, and n ≠ -1 .\int \left| \tan{ax} \right|dx =- (1)/(a) sgn (\tan{ax}) ln (\left|\cos{ax}\right|) + C when $ax\; \backslash in\; \backslash left(n\backslash pi\; -\; \backslash frac,\; n\backslash pi\; +\; \backslash frac\; \backslash right)$ for some integer .\int \left| \csc{ax} \right|dx =- (1)/(a) sgn (\csc{ax}) ln (\left| \csc{ax} + \cot{ax} \right|) + C when ax \in \left( n\pi, n\pi + \pi \right) for some integer .\int \left| \sec{ax} \right|dx = (1)/(a) sgn (\sec{ax}) ln (\left| \sec{ax} + \tan{ax} \right|) + C when $ax\; \backslash in\; \backslash left(n\backslash pi\; -\; \backslash frac,\; n\backslash pi\; +\; \backslash frac\; \backslash right)$ for some integer .\int \left| \cot{ax} \right|dx = (1)/(a) sgn (\cot{ax}) ln (\left|\sin{ax}\right|) + C when ax \in \left( n\pi, n\pi + \pi \right) for some integer .

If the function does not have any continuous antiderivative which takes the value zero at the zeros of (this is the case for the sine and the cosine functions), then is an antiderivative of on every interval on which is not zero, but may be discontinuous at the points where . For having a continuous antiderivative, 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, then we get:

\int \left| \sin{ax} \right|dx = { 2 \over a} \left\lfloor (ax)/(\pi) \right\rfloor - { 1 \over a} \cos{\left( ax - \left\lfloor (ax)/(\pi) \right\rfloor \pi \right)} + C \int \left|\cos {ax}\right|dx = { 2 \over a} \left\lfloor (ax)/(\pi) + (12)/(\right\rfloor) + { 1 \over a} \sin{\left( ax - \left\lfloor (ax)/(\pi) + (12)/(\right\rfloor) \pi \right)} + C

Special functions

, : Trigonometric integrals, : Exponential integral, : Logarithmic integral function, : Error function

\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 (\operatorname{li)/((x)}{x}dx) = ln x \operatorname{li}(x) - x \int \operatorname{erf}(x) dx = (e ^{- x ^{2}})/(\sqrt{\pi) }+x \operatorname(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 ^{inf ty} _{0} \sqrt{x}e ^{-x} dx = (1)/(2) \sqrt \pi (see also Gamma function)\int ^{inf ty} _{0} e ^{- a x ^{2}} dx = (1)/(2) \sqrt (\pi)/(a) for (the Gaussian integral)\int ^{inf ty{x} _{0} ^{2} e ^{- a x ^{2}} dx} = (1)/(4) \sqrt (\pi)/(a ^{3}) for \int ^{inf ty} _{0} x ^{2n} e ^{- a x ^{2}} dx = (2 n -1)/(2 a) \int ^{inf ty} _{0} x ^{2(n-1)} e ^{- a x ^{2}} dx = (( 2 n -1 )!!)/(\sqrt{ 2 ^{n+1}) (\pi)/(a ^{2n+1}) }= \frac \sqrtfor, is a positive integer and is the double factorial.\int ^{inf ty{x} _{0} ^{3} e ^{- a x ^{2}} dx} = (1)/(2 a ^{2}) when \int ^{inf ty} _{0} x ^{2n+1} e ^{- a x ^{2}} dx = (n)/(a) \int ^{inf ty} _{0} x ^{2n-1} e ^{- a x ^{2}} dx = (n!)/(2 a ^{n+1}) for, \int ^{inf ty} _{0} (x)/(e ^{x-1}) dx = (\pi ^{2})/(6) (see also Bernoulli number)\int ^{inf ty} _{0} (x ^{2})/(e ^{x-1}) dx = 2 \zeta( 3 ) ≈ 2 . 40 \int ^{inf ty} _{0} (x ^{3})/(e ^{x-1}) dx = (\pi ^{4})/(15) \int ^{inf ty} _{0} (\sin{x)/() }\,dx = \frac (see sinc function and the Dirichlet integral)\int ^{inf ty (\sin ^{2{x})/()} _{0} }\,dx = \frac\int ^{(\pi)/(2)} _{0} \sin ^{n} xdx = \int ^{(\pi)/(2)} _{0} \cos ^{n} xdx = ((n -1 )!!)/(n!!) x \begin{cases} 1 & if n isodd \\ (\pi)/(2) & if n iseven. \end{cases} (if is a positive integer and !! is the double factorial).\int ^{\pi} _{- \pi} \cos(\alpha x)\cos ^{n(\beta} x) dx = \begin{cases} (2 \pi)/(2 ^{n}) \binom{n}{m} & |\alpha| = |\beta ( 2 m - n)| \\ 0 & otherwise \end{cases} (for integers with and, see also Binomial coefficient)\int ^{t} _{- t} \sin ^{m(\alpha} x) \cos ^{n(\beta} x) dx = 0 (for real, a non-negative integer, and an odd, positive integer; since the integrand is odd)\int ^{\pi} _{- \pi} \sin(\alpha x) \sin ^{n(\beta} x) dx = \begin{cases} ( -1 ) ^{\left( (n +1)/(\right) 2)} ( -1 ) ^{m} (2 \pi)/(2 ^{n}) \binom{n}{m} & n odd , \alpha = \beta ( 2 m - n) \\ 0 & otherwise \end{cases} (for integers with and, see also Binomial coefficient)\int ^{\pi} _{- \pi} \cos(\alpha x) \sin ^{n(\beta} x) dx = \begin{cases} ( -1 ) ^{\left( (n)/(\right) 2)} ( -1 ) ^{m} (2 \pi)/(2 ^{n}) \binom{n}{m} & n even , |\alpha| = |\beta ( 2 m - n)| \\ 0 & otherwise \end{cases} (for integers with and, see also Binomial coefficient)\int ^{inf ty} _{-inf ty} e ^{- (ax ^{2+bx+c)}} dx = \sqrt{ (\pi)/(a) }\exp\left[\frac{b^2-4ac}{4a}\right](where is the exponential function, and .)\int ^{inf ty} _{0} x ^{z-1} e ^{-x} dx = \Gamma(z) (where \Gamma(z) is the Gamma function)\int ^{1} _{0} \left( ln (1)/(x) \right) ^{pdx} = \Gamma(p +1 ) \int ^{1} _{0} x ^{\alpha-1} ( 1- x) ^{\beta-1} dx = (\Gamma(\alpha)\Gamma(\beta))/(\Gamma(\alpha + \beta)) (for and, see Beta function)\int ^{2 \pi} _{0} e ^{x} d \theta = 2 \pi I _{0} (x) (where is the modified Bessel function of the first kind)\int ^{2 \pi} _{0} e ^{x} d \theta = 2 \pi I _{0} \left(\sqrt{x ^{2} + y ^{2}\right)} \int ^{inf ty} _{-inf ty} \left( 1 + (x ^{2})/(\nu) \right) ^{- (\nu +1)/(2)} dx = (\sqrt{\nu \pi)/(\Gamma\left( (\nu)/(2)) \right)} {\Gamma\left( (\nu +1)/(2) \right)} (for, this is related to the probability density function of Student's t-distribution)

If the function has bounded variation on the interval, then the method of exhaustion provides a formula for the integral:$$\backslash int\_a^b\; =\; (b\; -\; a)\; \backslash sum\backslash limits\_^\backslash infty\; 2^\; f(a\; +\; m\backslash left(\backslash right)2^).$$

The "sophomore's dream":attributed to Johann Bernoulli.

Further reading

• • Book: Taschenbuch der Mathematik .

    de:Taschenbuch der Mathematik

    . Ilja Nikolaevič . Bronstein . Konstantin Adolfovič . Semendjajew . Günter . Grosche . Viktor . Ziegler . Dorothea . Ziegler . Weiß, Jürgen . Viktor . Ziegler . 1 . 1987 . 23 . 1945 . Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig) . Thun and Frankfurt am Main . German . 3-87144-492-8.
• Book: Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Daniel . Zwillinger . Victor Hugo . Moll . Scripta Technica, Inc. . Table of Integrals, Series, and Products . . 2015 . October 2014 . 8 . English . 978-0-12-384933-5 . 2014010276 . 2016-02-21--> . Gradshteyn and Ryzhik. (Several previous editions as well.)
• Book: Anatolii Platonovich (Прудников, Анатолий Платонович) . Prudnikov . Anatolii Platonovich Prudnikov . Yuri A. (Брычков, Ю. А.) . Brychkov. Yuri A. Brychkov--> . Oleg Igorevich (Маричев, Олег Игоревич) . Marichev . Oleg Igorevich Marichev . Integrals and Series . 1 . English . N. M. . Queen . 1–5 . . 1981−1986 (Russian) . 1988–1992 . 2-88124-097-6. . Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yuri 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, / 9781584889564.
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. . (Many earlier editions as well.)
• , Integraltafeln oder Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810)
• , Integral Tables Or A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of ''Integraltafeln'']
• David 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

• Paul's Online Math Notes
• A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
• Math Major: A Table of Integrals
• Web site: O'Brien . Francis J. Jr. . 500 Integrals of Elementary and Special Functions. Derived integrals of exponential, logarithmic functions and special functions.
• Rule-based Integration Precisely defined indefinite integration rules covering a wide class of integrands
• Richard J. . Mathar . Yet another table of integrals . 1207.5845 . 2012. math.CA .

Derivations

• Victor Hugo Moll, The Integrals in Gradshteyn and Ryzhik

Online service

• Integration examples for Wolfram Alpha

Open source programs

• wxmaxima gui for Symbolic and numeric resolution of many mathematical problems

Videos

• The Single Most Overpowered Integration Technique in Existence. YouTube Video by Flammable Maths on symmetries

Notes and References

1. Book: Hirsch, Meyer . Integraltafeln: oder, Sammlung von integralformeln . 1810 . Duncker & Humblot . de.
2. [Serge Lang]
3. "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012