Lists of integrals
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Calculus  



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Integration is the basic operation in integral 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 Meier Hirsch (aka 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 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 closedform expressions have closedform 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 e^{−x2}, 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 Gfunction.
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, 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 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.
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 not in general necessary. For instance in
there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above would 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 −iπ when using a path above the origin and iπ 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:
Rational functions
 More integrals: List of integrals of rational functions
These rational functions have a nonintegrable singularity at 0 for a ≤ −1.
 (Cavalieri's quadrature formula)

 More generally,^{[1]}
Exponential functions
 More integrals: List of integrals of exponential functions
Logarithms
 More integrals: List of integrals of logarithmic functions
Trigonometric functions
 More integrals: List of integrals of trigonometric functions

 (See Integral of the secant function. This result was a wellknown conjecture in the 17th century.)

 (see integral of secant cubed)
Inverse trigonometric functions
 More integrals: List of integrals of inverse trigonometric functions
Hyperbolic functions
 More integrals: List of integrals of hyperbolic functions
Inverse hyperbolic functions
 More integrals: List of integrals of inverse hyperbolic functions
Products of functions proportional to their second derivatives
Absolutevalue functions
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
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):
when for some integer n.
when for some integer n.
when for some integer n.
when for some integer n.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. 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 π, then we get:
Special functions
Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function
Definite integrals lacking closedform 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.
 (see also Gamma function)
 for a > 0 (the Gaussian integral)
 for a > 0
 for a > 0, n is 1, 2, 3, ... and !! is the double factorial.
 when a > 0
 for a > 0, n = 0, 1, 2, ....
 (see also Bernoulli number)
 (see sinc function and Sine integral)
 (if n is an even integer and n ≥ 2)
 (if n is an odd integer and n ≥ 3)
 (for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient)
 (for α, β real and n nonnegative integer, see also Symmetry)
 (for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient)
 (for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient)
 (where exp[u] is the exponential function e^{u}, and a > 0)
 (where is the Gamma function)
 (for Re(α) > 0 and Re(β) > 0, see Beta function)
 (where I_{0}(x) is the modified Bessel function of the first kind)
 (for ν > 0 , this is related to the probability density function of Student's tdistribution)
If the function f has bounded variation on the interval [a,b], then the method of exhaustion provides a formula for the integral:
The change of variable gives
and, under this form the result appears in List of integrals of exponential functions#Definite integrals
The "sophomore's dream"
attributed to Johann Bernoulli.
See also
 Incomplete gamma function
 Indefinite sum
 List of limits
 List of mathematical series
 Symbolic integration
References
 ↑ "Reader Survey: logx + C", Tom Leinster, The ncategory Café, March 19, 2012
Further reading
 Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 0486612724. LCCN 6460036. MR 0167642. ISBN 9780486612720. LCCN 6512253.
 Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea, eds. Taschenbuch der Mathematik (in German). 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). ISBN 3871444928.
 Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo, eds. Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 0123849330. LCCN 2014010276. ISBN 9780123849335. (Several previous editions as well.)
 Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yuri A. (Брычков, Ю. А.); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (1988–1992) [1981−1986 (Russian)]. Integrals and Series. 1–5. Translated by Queen, N. M. (1 ed.). (Nauka) Gordon & Breach Science Publishers/CRC Press. ISBN 2881240976.. Second revised edition (Russian), volume 1–3, FizikoMatematicheskaya Literatura, 2003.
 Yuri A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, FizikoMatematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 158488956X / 9781584889564.
 Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1584882913. (Many earlier editions as well.)
 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 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
 O'Brien, Francis J. Jr. "500 Integrals". Derived integrals of exponential, logarithmic functions and special functions.
 Rulebased Mathematics Precisely defined indefinite integration rules covering a wide class of integrands
 Mathar, Richard J. (2012). "Yet another table of integrals". arXiv:1207.5845.