Lists of integrals


Lists of integrals

See the following pages for 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 arc hyperbolic functions
* List of integrals of exponential functions
* List of integrals of logarithmic functions

Tables of integrals

Integration is one of the two basic operations in 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.

"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 of simple functions

Irrational functions

:"more integrals: List of integrals of irrational functions":int {dx over sqrt{a^2-x^2 = sin^{-1} {x over a} + C:int {-dx over sqrt{a^2-x^2 = cos^{-1} {x over a} + C:int {dx over x sqrt{x^2-a^2 = {1 over a} sec^{-1} + C:int sec{x} , dx = ln{left| sec{x} + an{x} ight + C:int csc{x} , dx = -ln{left| csc{x} + cot{x} ight + C:int sec^2 x , dx = an x + C:int csc^2 x , dx = -cot x + C:int sec{x} , an{x} , dx = sec{x} + C:int csc{x} , cot{x} , dx = -csc{x} + C:int sin^2 x , dx = frac{1}{2}(x - frac{sin 2x}{2} ) + C = frac{1}{2}(x - sin xcos x ) + C :int cos^2 x , dx = frac{1}{2}(x + frac{sin 2x}{2}) + C = frac{1}{2}(x + sin xcos x ) + C :int sec^3 x , dx = frac{1}{2}sec x an x + frac{1}{2}ln|sec x + an x| + C:: (see integral of secant cubed):int sin^n x , dx = - frac{sin^{n-1} {x} cos {x{n} + frac{n-1}{n} int sin^{n-2}{x} , dx:int cos^n x , dx = frac{cos^{n-1} {x} sin {x{n} + frac{n-1}{n} int cos^{n-2}{x} , dx:int arctan{x} , dx = x , arctan{x} - frac{1}{2} ln{left| 1 + x^2 ight + C

Hyperbolic functions

:"more integrals: List of integrals of hyperbolic functions":int sinh x , dx = cosh x + C:int cosh x , dx = sinh x + C:int anh x , dx = ln| cosh x | + C:int mbox{csch},x , dx = lnleft| anh {x over2} ight| + C:int mbox{sech},x , dx = arctan(sinh x) + C:int coth x , dx = ln| sinh x | + C:int mbox{sech}^2 x, dx = anh x + C

Inverse hyperbolic functions

: int operatorname{arcsinh}, x , dx = x, operatorname{arcsinh}, x - sqrt{x^2+1} + C: int operatorname{arccosh}, x , dx = x, operatorname{arccosh}, x - sqrt{x^2-1} + C: int operatorname{arctanh}, x , dx = x, operatorname{arctanh}, x + frac{1}{2}log{(1-x^2)} + C: int operatorname{arccsch},x , dx = x, operatorname{arccsch}, x+ log{left [xleft(sqrt{1+frac{1}{x^2 + 1 ight) ight] } + C: int operatorname{arcsech},x , dx = x, operatorname{arcsech}, x- arctan{left(frac{x}{x-1}sqrt{frac{1-x}{1+x ight)} + C: int operatorname{arccoth},x , dx = x, operatorname{arccoth}, x+ frac{1}{2}log{(x^2-1)} + C

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_0^infty{sqrt{x},e^{-x},dx} = frac{1}{2}sqrt pi (see also Gamma function)

:int_0^infty{e^{-x^2},dx} = frac{1}{2}sqrt pi (the Gaussian integral)

:int_0^infty{frac{x}{e^x-1},dx} = frac{pi^2}{6} (see also Bernoulli number)

:int_0^infty{frac{x^3}{e^x-1},dx} = frac{pi^4}{15}

:int_0^inftyfrac{sin(x)}{x},dx=frac{pi}{2}

:int_0^frac{pi}{2}sin^n{x},dx=int_0^frac{pi}{2}cos^n{x},dx=frac{1 cdot 3 cdot 5 cdot cdots cdot (n-1)}{2 cdot 4 cdot 6 cdot cdots cdot n}frac{pi}{2} (if "n" is an even integer and scriptstyle{n ge 2})

:int_0^frac{pi}{2}sin^n{x},dx=int_0^frac{pi}{2}cos^n{x},dx=frac{2 cdot 4 cdot 6 cdot cdots cdot (n-1)}{3 cdot 5 cdot 7 cdot cdots cdot n} (if scriptstyle{n} is an odd integer and scriptstyle{n ge 3} )

:int_0^inftyfrac{sin^2{x{x^2},dx=frac{pi}{2}

:int_0^infty x^{z-1},e^{-x},dx = Gamma(z) (where Gamma(z) is the Gamma function)

:int_{-infty}^infty e^{-(ax^2+bx+c)},dx=sqrt{frac{pi}{aexpleft [frac{b^2-4ac}{4a} ight] (where exp [u] is the exponential function e^u, and a>0)

:int_{0}^{2 pi} e^{x cos heta} d heta = 2 pi I_{0}(x) (where I_{0}(x) is the modified Bessel function of the first kind)

:int_{0}^{2 pi} e^{x cos heta + y sin heta} d heta = 2 pi I_{0} left(sqrt{x^2 + y^2} ight)

:int_{-infty}^{infty}{(1 + x^2/ u)^{-( u + 1)/2}dx} = frac { sqrt{ u pi} Gamma( u/2)} {Gamma(( u + 1)/2))},, u > 0,, 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:

:int_a^b{f(x),dx} = (b - a) sumlimits_{n = 1}^infty {sumlimits_{m = 1}^{2^n - 1} {left( { - 1} ight)^{m + 1} } } 2^{ - n} f(a + mleft( {b - a} ight)2^{-n} ).

The "sophomore's dream"

:egin{align}int_0^1 x^{-x},dx &= sum_{n=1}^infty n^{-n} &&(= 1.29dots)\int_0^1 x^x ,dx &= sum_{n=1}^infty -(-1)^nn^{-n} &&(= 0.783430510712dots)end{align}

attributed to Johann Bernoulli; see sophomore's dream

Historical development of integrals

A compilation of a list of integrals (Integraltafeln) and techniques of integral calculuswas 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 ofthe 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.Since 1968 there is the Risch algorithm for determining indefinite integrals.

Other lists of integrals

Gradshteyn and Ryzhik contains a large collection of results. Other useful resources include the "CRC Standard Mathematical Tables and Formulae" and Abramowitz and Stegun. A&S contains many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. There are several web sites which have tables of integrals and integrals on demand.

References

* Besavilla: Engineering Review Center, "Engineering Mathematics (Formulas)", Mini Booklet

* Milton Abramowitz and Irene A. Stegun, eds. "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. [http://www.mathtable.com/gr Errata.] "(Several previous editions as well.)"

* 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.)"

See also

* List of mathematical series

External links

Tables of integrals

* [http://www.sosmath.com/tables/tables.html S.O.S. Mathematics: Tables and Formulas]
* [http://tutorial.math.lamar.edu/pdf/Common_Derivatives_Integrals.pdf Paul's Online Math Notes]

Historical

* Meyer Hirsch, [http://books.google.com/books?id=Cdg2AAAAMAAJ Integraltafeln, oder, Sammlung von Integralformeln] (Duncker un Humblot, Berlin, 1810)
* Meyer Hirsch, [http://books.google.com/books?id=NsI2AAAAMAAJ Integral Tables, Or, A Collection of Integral Formulae] (Baynes and son, London, 1823) [English translation of "Integraltafeln"]
* David de Bierens de Haan, [http://www.archive.org/details/nouvetaintegral00haanrich Nouvelles Tables d'Intégrales définies] (Engels, Leiden, 1862)
* Benjamin O. Pierce [http://books.google.com/books?id=pYMRAAAAYAAJ A short table of integrals - revised edition] (Ginn & co., Boston, 1899)


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Lists of mathematics topics — This article itemizes the various lists of mathematics topics. Some of these lists link to hundreds of articles; some link only to a few. The extremely long list of mathematics articles contains all mathematical articles in alphabetical order.… …   Wikipedia

  • List of integrals of trigonometric functions — The following is a list of integrals (antiderivative functions) of trigonometric functions. For integrals involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of Integral… …   Wikipedia

  • List of integrals of irrational functions — NOTOC The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. = Integrals involving r = sqrt{x^2+a^2} = : int r ;dx = frac{1}{2}left(x r… …   Wikipedia

  • List of integrals of rational functions — The following is a list of integrals (antiderivative functions) of rational functions. For a more complete list of integrals, see lists of integrals.:::: intfrac{1}{(ax^2+bx+c)^n} dx= frac{2ax+b}{(n 1)(4ac b^2)(ax^2+bx+c)^{n 1+frac{(2n 3)2a}{(n… …   Wikipedia

  • List of integrals of arc hyperbolic functions — The following is a list of integrals (antiderivative functions) of inverse hyperbolic functions. For a complete list of integral functions, see lists of integrals.: intmathrm{arsinh},frac{x}{c},dx = x,mathrm{arsinh},frac{x}{c} sqrt{x^2+c^2}:… …   Wikipedia

  • List of integrals of inverse trigonometric functions — The following is a list of integrals (antiderivative formulas) for integrands that contain inverse trigonometric functions (also known as arc functions ). For a complete list of integral formulas, see lists of integrals.Note: There are three… …   Wikipedia

  • List of derivatives and integrals in alternative calculi — This is a table of derivatives and integrals in alternative calculi. In the following table is the digamma function, is the K function, is subfactorial, are the generalized to real numbers Bernoulli p …   Wikipedia

  • Integral — This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation). A definite integral of a function can be represented as the signed area of the region bounded by its… …   Wikipedia

  • Calculus — This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables …   Wikipedia

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.