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| In [[calculus]], '''symbolic integration''' is the problem of finding a formula for the [[antiderivative]], or '''indefinite integral''', of a given function ''f''(''x''), i.e. to find the differentiable function ''F''(''x'') such that
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| :<math>\frac{dF}{dx} = f(x).</math>
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| This is also denoted
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| :<math>F(x) = \int f(x) \, dx.</math>
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| ==Discussion==
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| The term '''symbolic''' is used to distinguish this problem from that of [[numerical integration]], where the value of ''F'' at a particular input or set of inputs, rather than a general formula for ''F'', is sought.
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| Both problems were held to be of practical and theoretical importance long before the time of digital computers, but they are now generally considered the domain of [[computer science]], as computers are most often used currently to tackle individual instances.
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| Finding the derivative of an expression is a straightforward process for which it is easy to construct an [[algorithm]]. The reverse question of finding the integral is much more difficult. Many expressions which are relatively simple do not have integrals that can be expressed in [[Closed-form expression|closed form]]. See [[antiderivative]] for more details.
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| A procedure called the [[Risch algorithm]] exists which is capable of determining if the integral of an [[elementary function]] (function built from a finite number of [[exponential function|exponential]]s, [[logarithm]]s, [[coefficient|constant]]s, and [[nth roots]] through [[function composition|composition]] and combinations using the four [[arithmetic|elementary operations]]) is elementary and returning it if it does. In its original form, Risch algorithm was not suitable for a direct implementation, and its complete implementation took a long time. It was first implemented in [[Reduce (computer algebra system)|Reduce]] in the case of purely transcendental functions; the case of purely algebraic functions was solved and implemented in Reduce by [[James H. Davenport]]; the general case was solved and implemented in [[Axiom (computer algebra system)|Axiom]] by Manuel Bronstein.
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| However, the Risch algorithm applies only to ''indefinite'' integrals and most of the integrals of interest to physicists, theoretical chemists and engineers, are ''definite'' integrals often related to [[Laplace transforms]], [[Fourier transform]]s and [[mellin transform|Mellin transforms]]. Lacking of a general algorithm, the developers of [[computer algebra system]]s, have implemented [[heuristic (computer science)|heuristics]] based on pattern-matching and the exploitation of special functions, in particular the [[incomplete gamma function]]<ref>[[Keith Geddes|K.O Geddes]], M.L. Glasser, R.A. Moore and T.C. Scott, ''Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions'', AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149–165, [http://www.springerlink.com/content/t7571u653t83037j/] </ref> Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving many definite integrals encountered by practical engineering applications. Earlier systems such as [[Macsyma]] had a few definite integrals related to special functions within a look-up table. However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, [[pattern matching]] and other manipulations, was pioneered by developers of the [[Maple (software)|Maple]]<ref>K.O. Geddes and T.C. Scott, ''Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms'', Proceedings of the 1989 Computers and Mathematics conference, (held at MIT June 12, 1989), edited by E. Kaltofen and S.M. Watt, Springer-Verlag, New York, (1989), pp. 192–201. [http://portal.acm.org/citation.cfm?id=93094]</ref> system then later emulated by [[Mathematica]], [[Axiom (computer algebra system)|Axiom]], [[MuPAD]] and other systems.
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| == Example ==
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| For example:
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| :<math>\int x^2\,dx = \frac{x^3}{3} + C</math>
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| is a symbolic result for an indefinite integral (here C is a [[constant of integration]]),
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| :<math>\int_{-1}^1 x^2\,dx = \left[\frac{x^3}{3}\right]_{-1}^1= \frac{1^3}{3} - \frac{(-1)^3}{3}=\frac{2}{3}</math>
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| is a symbolic result for a definite integral, and
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| :<math>\int_{-1}^1 x^2\,dx \approx 0.6667</math>
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| is a numerical result for the same definite integral.
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| == See also ==
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| * [[Integral|Definite integral]]
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| * [[Elementary function]]
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| * [[Risch algorithm]]
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| * [[Symbolic computation]]
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| == References ==
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| <references/>
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| *{{Citation
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| |last= Bronstein
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| |first= Manuel
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| |title= Symbolic Integration 1 (transcendental functions)
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| |edition= 2
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| |year= 1997
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| |publisher= Springer-Verlag
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| |isbn= 3-540-60521-5
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| |doi=}}
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| *{{Citation
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| |last= Moses
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| |first= Joel
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| |author-link= Joel Moses
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| |title= Symbolic integration: the stormy decade
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| |journal= Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation
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| |pages= 427–440
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| |date= March 23–25, 1971
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| |location= Los Angeles, California
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| |doi=}}
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| == External links ==
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| * {{MathWorld|urlname=RischAlgorithm|title=Risch Algorithm|author=Bhatt, Bhuvanesh}}
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| * [http://integrals.wolfram.com Wolfram Integrator] — Free online symbolic integration with [[Mathematica]]
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| [[Category:Computer algebra]]
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| [[Category:Differential algebra]]
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