Integration
Integration may refer to:
Biology
* Multisensory integration
* Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technolo ...
is the basic operation in
integral calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
. While
differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product differentiation, in marketing
* Differentiated service, a service that varies with the identity ...
has straightforward
rules
Rule or ruling may refer to:
Education
* Royal University of Law and Economics (RULE), a university in Cambodia
Human activity
* The exercise of political or personal control by someone with authority or power
* Business rule, a rule pert ...
by which the derivative of a complicated
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
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
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s.
Historical development of integrals
A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician (aka ) 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
David Bierens de Haan (3 May 1822, in Amsterdam – 12 August 1895, in Leiden) was a Dutch mathematician and historian of science.
Biography
Bierens de Haan was a son of the rich merchant Abraham Pieterszoon de Haan (1795–1880) and Catharin ...
for his ''
Tables d'intégrales définies
David Bierens de Haan (3 May 1822, in Amsterdam – 12 August 1895, in Leiden) was a Dutch mathematician and historian of science.
Biography
Bierens de Haan was a son of the rich merchant Abraham Pieterszoon de Haan (1795–1880) and Catharina J ...
'', supplemented by ''
Supplément aux tables d'intégrales définies
David Bierens de Haan (3 May 1822, in Amsterdam – 12 August 1895, in Leiden) was a Dutch mathematician and historian of science.
Biography
Bierens de Haan was a son of the rich merchant Abraham Pieterszoon de Haan (1795–1880) and Catharina J ...
'' in ca. 1864. A new edition was published in 1867 under the title ''
Nouvelles tables d'intégrales définies
David Bierens de Haan (3 May 1822, in Amsterdam – 12 August 1895, in Leiden) was a Dutch mathematician and historian of science.
Biography
Bierens de Haan was a son of the rich merchant Abraham Pieterszoon de Haan (1795–1880) and Catharina J ...
''. 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
''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''Table of Integrals, Series, and Products ...
. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.
Not all
closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
s have closed-form antiderivatives; this study forms the subject of
differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.
Overview
Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fiel ...
, which was initially developed by
Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
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 , whose antiderivative is (up to constants) the
error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as:
:\operatorname z = \frac\int_0^z e^\,\mathrm dt.
This integral is a special (non- elementa ...
.
Since 1968 there is the
Risch algorithm
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra w ...
for determining indefinite integrals that can be expressed in term of
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
s, typically using a
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the
Meijer G-function
In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the M ...
.
Lists of integrals
More detail may be found on the following pages for the lists of
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s:
*
List of integrals of rational functions
The following is a list of integrals (antiderivative functions) of rational functions.
Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:
: \frac, and \frac.
which can th ...
*
List of integrals of irrational functions
The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration
In calculus, the constant of integra ...
*
List of integrals of trigonometric functions
The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antide ...
*
List of integrals of inverse trigonometric functions
The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals.
* The inverse trigonometric functions are also know ...
*
List of integrals of hyperbolic functions
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals.
In all formulas the constant ''a'' is assumed to be nonzero, and ''C''
denotes the constant ...
*
List of integrals of inverse hyperbolic functions
The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals.
* In all formulas the constant is assumed to be nonzer ...
*
List of integrals of exponential functions
The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals.
Indefinite integral
Indefinite integrals are antiderivative functions. A constant (the constant of inte ...
*
List of integrals of logarithmic functions
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals.
''Note:'' ''x'' > 0 is assumed throughout this article, and the constant of integratio ...
*
List of integrals of Gaussian functions
In the expressions in this article,
:\phi(x) = \frace^
is the standard normal probability density function,
:\Phi(x) = \int_^x \phi(t) \, dt = \frac\left(1 + \operatorname\left(\frac\right)\right)
is the corresponding cumulative distribution fu ...
Gradshteyn,
Ryzhik,
Geronimus,
Tseytlin, Jeffrey, Zwillinger, and
Moll's (GR) ''
Table of Integrals, Series, and Products
''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''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
Elementary may refer to:
Arts, entertainment, and media Music
* ''Elementary'' (Cindy Morgan album), 2001
* ''Elementary'' (The End album), 2007
* ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977
Other uses in arts, entertainment, a ...
and
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined ...
, volume 4–5 are tables of
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
s). 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
''Bronshtein and Semendyayev'' (often just ''Bronshtein'' or ''Bronstein'', sometimes ''BS'') is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas originally compiled by the Rus ...
's ''
Guide Book to Mathematics'', ''
Handbook of Mathematics
''Bronshtein and Semendyayev'' (often just ''Bronshtein'' or ''Bronstein'', sometimes ''BS'') is the informal name of a comprehensive handbook of fundamental working knowledge of mathematics and table of formulas originally compiled by the Rus ...
'' or ''
Users' Guide to Mathematics'', and other mathematical handbooks.
Other useful resources include
Abramowitz and Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and ...
and the
Bateman Manuscript Project
The Bateman Manuscript Project was a major effort at collation and encyclopedic compilation of the mathematical theory of special functions. It resulted in the eventual publication of five important reference volumes, under the editorship of Arth ...
. 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
WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data.
WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Math ...
can show results, and for some simpler expressions, also the intermediate steps of the integration.
Wolfram Research
Wolfram Research, Inc. ( ) is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include ...
also operates another online service, the Mathematica Online Integrator.
Integrals of simple functions
''C'' is used for an
arbitrary constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
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
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s.
These formulas only state in another form the assertions in the
table of derivatives
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Elementary rules of differentiation
Unless otherwise stated, all functions are functions of real numbers (R) that return re ...
.
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
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand ...
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
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
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:
Rational functions
*
The following function has a non-integrable singularity at 0 for :
*
(
Cavalieri's quadrature formula)
*
*
**More generally,
[Reader Survey: log, ''x'', + ''C'']
, Tom Leinster, ''The ''n''-category Café'', March 19, 2012
*
Exponential functions
*
*
*
*
*(if is a positive integer)
*(if is a positive integer)
Logarithms
*
*
Trigonometric functions
*
*
*
*
*
** (See Integral of the secant function. This result was a well-known conjecture in the 17th century.)
*
*
*
*
*
*
*
*
*
*
** (See integral of secant cubed
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
:\begin
\int \sec^3 x \, dx
&= \tfrac12\sec x \tan x + \tfrac12 \int \sec x\, dx + C \\ mu&= \tfrac12(\sec x \tan x + \ln \left, \sec x + \ta ...
.)
*
*
*
Inverse trigonometric functions
*
*
*
*
*
*
Hyperbolic functions
*
*
*
*
*
*
*
*
*
*
Inverse hyperbolic functions
*
*
*
*
*
*
Products of functions proportional to their second derivatives
*
*
*
*
Absolute-value functions
Let be a continuous function, that has at most one zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
. If has a zero, let be the unique antiderivative of that is zero at the root of ; otherwise, let be any antiderivative of . Then
where is the sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To a ...
, which takes the values −1, 0, 1 when 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 is here for insuring the continuity of the integral.
This gives the following formulas (where ), which are valid over any interval where is continuous (over larger intervals, the constant must be replaced by a piecewise constant
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only ...
function):
*when is odd, and .
*when for some integer .
*when for some integer .
*when for some integer .
*when for some integer .
If the function does not have any continuous antiderivative which takes the value zero at the zeros of (this is the case for the sine and the cosine functions), then is an antiderivative of on every interval on which is not zero, but may be discontinuous at the points where . For having a continuous antiderivative, one has thus to add a well chosen step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having on ...
. If we also use the fact that the absolute values of sine and cosine are periodic with period , then we get:
*
*
Special functions
, : Trigonometric integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operatorname(x) = -\int_x^\infty\frac\ ...
s, : Exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of ...
, : Logarithmic integral function
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
, : Error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as:
:\operatorname z = \frac\int_0^z e^\,\mathrm dt.
This integral is a special (non- elementa ...
*
*
*
*
*
*
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
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
)
* for (the Gaussian integral
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
\int_^\infty e^\,dx = \s ...
)
* for
*for , is a positive integer and is the double factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is,
:n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
For even , the ...
.
* when
*for ,
* (see also Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions ...
)
*
*
* (see sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the ...
and the Dirichlet integral
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
: \int_0^\inft ...
)
*
*(if is a positive integer and !! is the double factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is,
:n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
For even , the ...
).
*(for integers with and , see also Binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
)
*(for real, a non-negative integer, and an odd, positive integer; since the integrand is odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
)
*(for integers with and , see also Binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
)
*(for integers with and , see also Binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
)
*exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, and .)
*(where is the Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
)
*
*(for and , see Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^ ...
)
* (where is the modified Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the first kind)
*
*(for , this is related to the probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
of Student's ''t''-distribution)
If the function has bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
on the interval , then the method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area b ...
provides a formula for the integral:
The "sophomore's dream
In mathematics, the sophomore's dream is the pair of identities (especially the first)
:\begin
\int_0^1 x^\,dx &= \sum_^\infty n^ \\
\end
:\begin
\int_0^1 x^x \,dx &= \sum_^\infty (-1)^n^ = - \sum_^\infty (-n)^
\end
discovered in 1697 by Jo ...
":
attributed to Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating ...
.
See also
*
*
*
*
*
*
*
*
*
*
References
Further reading
*
*
* (Several previous editions as well.)
* . Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
* Yuri 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, / 9781584889564.
* Daniel Zwillinger. ''CRC Standard Mathematical Tables and Formulae'', 31st edition. Chapman & Hall/CRC Press, 2002. . ''(Many earlier editions as well.)''
* ,
Integraltafeln oder Sammlung von Integralformeln
' (Duncker und Humblot, Berlin, 1810)
* ,
Integral Tables Or A Collection of Integral Formulae
' (Baynes and son, London, 1823) nglish translation of ''Integraltafeln''* David Bierens de Haan
David Bierens de Haan (3 May 1822, in Amsterdam – 12 August 1895, in Leiden) was a Dutch mathematician and historian of science.
Biography
Bierens de Haan was a son of the rich merchant Abraham Pieterszoon de Haan (1795–1880) and Catharin ...
Nouvelles Tables d'Intégrales définies
(Engels, Leiden, 1862)
* Benjamin O. Pierc
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)
Math Major: A Table of Integrals
* Derived integrals of exponential, logarithmic functions and special functions.
Rule-based Integration
Precisely defined indefinite integration rules covering a wide class of integrands
*
Derivations
Online service
Open source programs
wxmaxima gui for Symbolic and numeric resolution of many mathematical problems
Videos
*
The Single Most Overpowered Integration Technique in Existence
'' YouTube Video by Flammable Maths on symmetries
{{DEFAULTSORT:Integrals
Integrals
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
Mathematical identities