List of Integrals Wiki

Lists of integrals 1

Lists of integralsIntegration is the basic operation in integral calculus. While differentiation has easy rules by which the derivative ofa complicated function can be found by differentiating its simpler component functions, integration does not, sotables of known integrals are often useful. This page lists some of the most common antiderivatives.

Historical development of integralsA compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the Germanmathematician Meyer Hirsch in 1810. These tables were republished in the United Kingdom in 1823. Moreextensive tables were compiled in 1858 by the Dutch mathematician David de Bierens de Haan. A new edition waspublished in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until themiddle 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 Galoistheory, 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 aclosed form antiderivative is ex2, 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 ofelementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementaryfunctions can be manipulated symbolically using general functions such as the Meijer G-function.

Lists of integralsMore 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 functionsGradshteyn, Ryzhik, Jeffrey, Zwillinger's Table of Integrals, Series, and Products contains a large collection ofresults. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (withvolumes 13 listing integrals and series of elementary and special functions, volume 45 are tables of Laplacetransforms). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's Tables of IndefiniteIntegrals, or as chapters in Zwillinger's CRC Standard Mathematical Tables and Formulae, Bronstein andSemendyayev'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 containmany identities concerning specific integrals, which are organized with the most relevant topic instead of beingcollected 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 [1].

Integrals of simple functionsC is used for an arbitrary constant of integration that can only be determined if something about the value of theintegral 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 singularityWhen there is a singularity in the function being integrated such that the integral becomes undefined, i.e., it is notLebesgue integrable, then C does not need to be the same on both sides of the singularity. The forms below normallyassume the Cauchy principal value around a singularity in the value of C but this is not in general necessary. Forinstance in

there is a singularity at 0 and the integral becomes infinite there. If the integral above was used to give a definiteintegral between -1 and 1 the answer would be 0. This however is only the value assuming the Cauchy principalvalue for the integral around the singularity. If the integration was done in the complex plane the result woulddepend on the path around the origin, in this case the singularity contributes i when using a path above the originand i for a path below the origin. A function on the real line could use a completely different value of C on eitherside of the origin as in:

Rational functionsmore integrals: List of integrals of rational functions

These rational functions have a non-integrable singularity at 0 for a 1.

(Cavalieri's quadrature formula)


Exponential functionsmore integrals: List of integrals of exponential functions

Logarithmsmore integrals: List of integrals of logarithmic functions

Trigonometric functionsmore integrals: List of integrals of trigonometric functions

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

(see integral of secant cubed)


Inverse trigonometric functionsmore integrals: List of integrals of inverse trigonometric functions

Hyperbolic functionsmore integrals: List of integrals of hyperbolic functions

Inverse hyperbolic functionsmore integrals: List of integrals of inverse hyperbolic functions


Products of functions proportional to their second derivatives

Absolute value functions

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


Definite integrals lacking closed-form antiderivativesThere are some functions whose antiderivatives cannot be expressed in closed form. However, the values of thedefinite integrals of some of these functions over some common intervals can be calculated. A few useful integralsare 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 )

(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:

The "sophomore's dream"

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 (.. ), I.M. Ryzhik (.. ); Alan Jeffrey, Daniel Zwillinger, editors. Table of

Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. [2]

(Several previous editions as well.)

A.P. Prudnikov (.. ), Yu.A. Brychkov (.. ), O.I. Marichev (.. ). Integralsand Series. First edition (Russian), volume 15, Nauka, 19811986. First edition (English, translated from theRussian by N.M. Queen), volume 15, Gordon & Breach Science Publishers/CRC Press, 19881992, ISBN2-88124-097-6. Second revised edition (Russian), volume 13, Fiziko-Matematicheskaya Literatura, 2003.

Yu.A. Brychkov (.. ), Handbook of Special Functions: Derivatives, Integrals, Series and OtherFormulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRCPress, 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 [3] (Duncker und Humblot, Berlin, 1810) Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae [4] (Baynes and son, London, 1823)

[English translation of Integraltafeln] David Bierens de Haan, Nouvelles Tables d'Intgrales dfinies [5] (Engels, Leiden, 1862) Benjamin O. Pierce A short table of integrals - revised edition [6] (Ginn & co., Boston, 1899)

External links

Tables of integrals S.O.S. Mathematics: Tables and Formulas [7] (warning: may serve popunders) Paul's Online Math Notes [8]

A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More ExoticFunctions): Indefinite Integrals [9] Definite Integrals [10]

Math Major: A Table of Integrals [11]

O'Brien, Francis J. Jr. Integrals [12] Derived integrals of exponential and logarithmic functions Rule-based Mathematics [13] Precisely defined indefinite integration rules covering a wide class of integrands

Derivations V. H. Moll, The Integrals in Gradshteyn and Ryzhik [14]

Online service Integration examples for Wolfram Alpha [15]

Open source programs wxmaxima gui for Symbolic and numeric resolution of many mathematical problems [16]

References[1] http:/ / integrals. wolfram. com/ index. jsp[2] http:/ / www. mathtable. com/ gr[3] http:/ / books. google. com/ books?id=Cdg2AAAAMAAJ[4] http:/ / books. google. com/ books?id=NsI2AAAAMAAJ[5] http:/ / www. archive. org/ details/ nouvetaintegral00haanrich[6] http:/ / books. google. com/ books?id=pYMRAAAAYAAJ[7] http:/ / www. sosmath. com/ tables/ tables. html[8] http:/ / tutorial. math. lamar. edu/ pdf/ Common_Derivatives_Integrals. pdf[9] http:/ / pi. physik. uni-bonn. de/ ~dieckman/ IntegralsIndefinite/ IndefInt. html[10] http:/ / pi. physik. uni-bonn. de/ ~dieckman/ IntegralsDefinite/ DefInt. html[11] http:/ / mathmajor. org/ calculus-and-analysis/ table-of-integrals/[12] http:/ / www. docstoc. com/ docs/ 23969109/ 500-Integrals-of-Elementary-and-Special-Functions''500[13] http:/ / www. apmaths. uwo. ca/ RuleBasedMathematics/ index. html[14] http:/ / www. math. tulane. edu/ ~vhm/ Table. html[15] http:/ / www. wolframalpha. com/ examples/ Integrals. html[16] http:/ / wxmaxima. sourceforge. net/ wiki/ index. php/ Main_Page

Article Sources and Contributors 9

Article Sources and ContributorsLists of integrals Source: http://en.wikipedia.org/w/index.php?oldid=484002704 Contributors: 00Ragora00, Akikidis, Albert D. Rich, Amazins490, AngrySaki, Ant314159265,ArnoldReinhold, Asmeurer, BANZ111, BananaFiend, BehzadAhmadi, Bilboq, Bruno3469, Brutha, CWenger, Ciphers, Ccero, DJPhoenix719, DavidWBrooks, Dcirovic, Deineka, DerHexer,Dmcq, Doctormatt, Dogcow, Doraemonpaul, Dpb2104, Drahmedov, Dysprosia, Euty, FerrousTigrus, Fieldday-sunday, Fredrik, Giftlite, Giulio.orru, Gloriphobia, Happy-melon, IDGC, Icairns,Imperial Monarch, Itai, Itu, Ivan tambuk, JNW, Jaisenberg, Jimp, Jj137, John Vandenberg, Jon R W, Jumpythehat, Jwillbur, KSmrq, Kantorghor, Kilonum, Kusluj, LachlanA, LeaveSleaves,Legendre17, Lesonyrra, Linas, LizardJr8, Lzur, Macrakis, MathFacts, Michael Hardy, MrOllie, Msablic, Muro de Aguas, NNemec, Nbarth, New Math, NewEnglandYankee, NickFr, NinjaCross,Oleg Alexandrov, Perelaar, Phatsphere, Physman, Physmanir, Pimvantend, Pokipsy76, Pschemp, Qmtead, RobHar, Salih, Salix alba, Schneelocke, Scythe33, ShakataGaNai, Sseyler, Stpasha,TStein, TakuyaMurata, Template namespace initialisation script, Tetzcatlipoca, The Transhumanist, Thenub314, Tkreuz, Unyoyega, VasilievVV, Vedantm, Waabu, Wile E. Heresiarch,Willking1979, Woohookitty, Xanthoxyl, Yeungchunk, Ylai, Zmoney918, 263 anonymous edits

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Lists of integralsHistorical development of integralsLists of integralsIntegrals of simple functionsIntegrals with a singularityRational functionsExponential functionsLogarithmsTrigonometric functionsInverse trigonometric functionsHyperbolic functionsInverse hyperbolic functionsProducts of functions proportional to their second derivativesAbsolute value functionsSpecial functions

Definite integrals lacking closed-form antiderivativesThe "sophomore's dream"

References Historical

External links Tables of integrals Derivations Online service Open source programs

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