Lists of integralsFrom Wikipedia, the free encyclopedia

This article is mainly about indefinite integrals in calculus. For a list of definite integrals see List of definite integrals.

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.

Contents

■ 1 Historical development of integrals■ 2 Lists of integrals■ 3 Integrals of simple functions

■ 3.1 Integrals with a singularity■ 3.2 Rational functions■ 3.3 Exponential functions■ 3.4 Logarithms■ 3.5 Trigonometric functions■ 3.6 Inverse trigonometric functions■ 3.7 Hyperbolic functions■ 3.8 Inverse hyperbolic functions■ 3.9 Products of functions proportional to their second derivatives■ 3.10 Absolute value functions■ 3.11 Special functions

■ 4 Definite integrals lacking closed-form antiderivatives■ 4.1 The "sophomore's dream"

■ 5 See also■ 6 References

■ 6.1 Historical■ 7 External links

■ 7.1 Tables of integrals■ 7.2 Derivations■ 7.3 Online service■ 7.4 Open source programs

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.

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 theoremwhich 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 G-function.

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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, 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 (http://integrals.wolfram.com/index.jsp) .

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 integral becomes undefined, i.e., it is not Lebesgue integrable, 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 integral becomes infinite there. If the integral above was used to give a definite integral between -1 and 1 the answer would be 0. This however is only the value assuming the Cauchy principal value for the integral around the singularity. If the integration was done in the complex plane the result would depend on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπfor 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:

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Rational functions

more integrals: List of integrals of rational functions

These rational functions have a non-integrable 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 well-known conjecture in the 17th century.)

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(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

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Products of functions proportional to their second derivatives

Absolute value functions

Special functions

Ci, Si: Trigonometric integrals, Ei: Exponential integral, li: Logarithmic integral function, erf: Error function

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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.

(see also Gamma function)

(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 )

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(if is an odd integer and )

(for integers with and

, see also Binomial coefficient)

(for real and non-negative integer, see also Symmetry)

(for

integers with and , see also Binomial coefficient)

(for

integers with and , see also Binomial coefficient)

(where is the exponential function , and

)

(where is the Gamma function)

(the Beta function)

(where is the modified Bessel function of the first kind)

, 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:

Start by using the substitution

This brings the integral to the general form

(Click "show" at right to see a proof or "hide" to hide it.)

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which after integration by parts yields

and provided the first term vanishes at the end points, we get the recurrence relation

which upon computation gives

Applying to our integral, we notice that

Hence the final answer is:

The "sophomore's dream"

attributed to Johann Bernoulli.

See also

■ Indefinite sum■ Incomplete Gamma function■ List of mathematical series■ List of limits■ Symbolic integration

References1. ^ "Reader Survey: log|x| + C (http://golem.ph.utexas.edu/category/2012/03/reader_survey_logx_c.html) ", Tom Leinster,

The n-category Café, March 19, 2012

■ 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. (http://www.mathtable.com/gr) (Several previous editions as well.)

■ A.P. Prudnikov (А.П. Прудников), Yu.A. Brychkov (Ю.А. Брычков), O.I. Marichev (О.И. Маричев). Integrals 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,

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ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.

■ Yu.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, ISBN 1-58488-956-X.

■ 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 (http://books.google.com/books?id=Cdg2AAAAMAAJ) (Duncker und Humblot, Berlin, 1810)

■ Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (http://books.google.com/books?id=NsI2AAAAMAAJ) (Baynes and son, London, 1823) [English translation of Integraltafeln]

■ David Bierens de Haan, Nouvelles Tables d'Intégrales définies(http://www.archive.org/details/nouvetaintegral00haanrich) (Engels, Leiden, 1862)

■ Benjamin O. Pierce A short table of integrals - revised edition (http://books.google.com/books?id=pYMRAAAAYAAJ) (Ginn & co., Boston, 1899)

External links

Tables of integrals

■ Paul's Online Math Notes (http://tutorial.math.lamar.edu/pdf/Common_Derivatives_Integrals.pdf) ■ A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic

Functions): Indefinite Integrals (http://pi.physik.uni-bonn.de/~dieckman/IntegralsIndefinite/IndefInt.html) Definite Integrals (http://pi.physik.uni-bonn.de/~dieckman/IntegralsDefinite/DefInt.html)

■ Math Major: A Table of Integrals (http://mathmajor.org/calculus-and-analysis/table-of-integrals/) ■ O'Brien, Francis J. Jr. 500 Integrals (http://www.docstoc.com/docs/23969109/500-Integrals-of-Elementary-and

-Special-Functions) Derived integrals of exponential and logarithmic functions■ Rule-based Mathematics (http://www.apmaths.uwo.ca/RuleBasedMathematics/index.html) Precisely defined

indefinite integration rules covering a wide class of integrands

Derivations

■ V. H. Moll, The Integrals in Gradshteyn and Ryzhik (http://www.math.tulane.edu/~vhm/Table.html)

Online service

■ Integration examples for Wolfram Alpha (http://www.wolframalpha.com/examples/Integrals.html)

Open source programs

■ wxmaxima gui for Symbolic and numeric resolution of many mathematical problems(http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page)

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