Lists of Integrals
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Lists of Integrals

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 Meier Hirsch [de] (aka Meyer Hirsch [de]) 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 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 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.

Lists of integrals

More detail may be found on the following pages for the lists of integrals:

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 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:[1]

Rational functions

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

  • (Cavalieri's quadrature formula)
    • More generally,[2]

Exponential functions

Logarithms

Trigonometric functions

Inverse trigonometric functions

Hyperbolic functions

Inverse hyperbolic functions

Products of functions proportional to their second derivatives

Absolute-value functions

Let f be a continuous function, that has at most one zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then

where sgn(x) is the sign function, which takes the values -1, 0, 1 when x 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 g is here for insuring the continuity of the integral.

This gives the following formulas (where a ? 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a piecewise constant function):

  • when n is odd, and .
  • 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 ?, then we get:

  • []
  • []

Special functions

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

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)
  • for a > 0 (the Gaussian integral)
  • for a > 0
  • for a > 0, n is a positive integer and !! is the double factorial.
  • when a > 0
  • for a > 0, n = 0, 1, 2, ....
  • (see also Bernoulli number)
  • (see sinc function and the Dirichlet integral)
  • (if n is a positive integer and !! is the double factorial).
  • (for ?, ?, m, n integers with ? ? 0 and m, n >= 0, see also Binomial coefficient)
  • (for ?, ? real, n a non-negative integer, and m an odd, positive integer; since the integrand is odd)
  • (for ?, ?, m, n integers with ? ? 0 and m, n >= 0, see also Binomial coefficient)
  • (for ?, ?, m, n integers with ? ? 0 and m, n >= 0, see also Binomial coefficient)
  • (where exp[u] is the exponential function eu, and a > 0.)
  • (where is the Gamma function)
  • (for Re(?) > 0 and Re(?) > 0, see Beta function)
  • (where I0(x) is the modified Bessel function of the first kind)
  • (for ? > 0 , this is related to the probability density function of Student's t-distribution)

If the function f has bounded variation on the interval [a,b], then the method of exhaustion provides a formula for the integral:

The "sophomore's dream":

attributed to Johann Bernoulli.

See also

References

  1. ^ Serge Lang . A First Course in Calculus, 5th edition, p. 290
  2. ^ "Reader Survey: log|x| + C", Tom Leinster, The n-category 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.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  • Bronstein, Ilja Nikolaevi?; Semendjajew, Konstantin Adolfovi? (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 3-87144-492-8.
  • 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 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)
  • Prudnikov, Anatolii Platonovich (, ?); Brychkov, Yuri A. (?, ?. ?.); Marichev, Oleg Igorevich (?, ? ) (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 2-88124-097-6.. Second revised edition (Russian), volume 1-3, Fiziko-Matematicheskaya Literatura, 2003.
  • Yuri A. Brychkov (?. ?. ?), 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 / 9781584889564.
  • 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.)
  • Meyer Hirsch [de], Integraltafeln oder Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810)
  • Meyer Hirsch [de], 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

Derivations

Online service

Open source programs

Videos


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