Symbolic integration
Encyclopedia
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
This is also denoted
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.
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.
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
. See antiderivative
for more details.
A procedure called the Risch algorithm
exists which is capable of determining if an integral exists and returning it if it does, for many classes of expressions. Such algorithms are still being expanded.
However, the Risch algorithm applies 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 transforms
. An alternative to the Risch algorithm involves the combination of a computer algebra system
, pattern-matching and the exploitation of special functions, in particular the Incomplete gamma function
Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving definite integrals, in particular those encountered by practical engineering applications. This method was pioneered by developers of the Maple
system then later emulated by Mathematica
, MuPAD
and other systems. This approach has applications in experimental mathematics
.
is a symbolic result for an indefinite integral (here C is a constant of integration),
is a symbolic result for a definite integral, and
is a numerical result for the same definite integral.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
symbolic integration is the problem of finding a formula for the antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
, or indefinite integral, of a given function f(x), i.e. to find the differentiable function F(x) such that
This is also denoted
The term symbolic is used to distinguish this problem from that of numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
, where the value of F at a particular input or set of inputs, rather than a general formula for F, is sought.
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
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, as computers are most often used currently to tackle individual instances.
Finding the derivative of an expression is a straightforward process for which it is easy to construct an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
. 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
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
. See antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
for more details.
A procedure called the Risch algorithm
Risch algorithm
The Risch algorithm, named after Robert Henry Risch, is an algorithm for the calculus operation of indefinite integration . The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational...
exists which is capable of determining if an integral exists and returning it if it does, for many classes of expressions. Such algorithms are still being expanded.
However, the Risch algorithm applies 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
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s and Mellin transforms
Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
. An alternative to the Risch algorithm involves the combination of a computer algebra system
Computer algebra system
A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
, pattern-matching and the exploitation of special functions, in particular the Incomplete gamma function
Incomplete gamma function
In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of...
Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving definite integrals, in particular those encountered by practical engineering applications. This method was pioneered by developers of the Maple
Maple (software)
Maple is a general-purpose commercial computer algebra system. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada....
system then later emulated by Mathematica
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
, MuPAD
MuPAD
MuPAD is a computer algebra system . Originally developed by the MuPAD research group at the University of Paderborn, Germany, development was taken over by the company SciFace Software GmbH & Co...
and other systems. This approach has applications in experimental mathematics
Experimental mathematics
Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns...
.
Example
For example:is a symbolic result for an indefinite integral (here C is a constant of integration),
is a symbolic result for a definite integral, and
is a numerical result for the same definite integral.
See also
- AntiderivativeAntiderivativeIn calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
- Definite integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
- Elementary function
- Risch algorithmRisch algorithmThe Risch algorithm, named after Robert Henry Risch, is an algorithm for the calculus operation of indefinite integration . The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational...
External links
- Wolfram Integrator — Free online symbolic integration with MathematicaMathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
- Mathematical Assistant on Web — symbolic computations online. Allows to integrate in small steps (with hints for next step (integration by parts, substitution, partial fractions, application of formulas and others), powered by Maxima
- Online integral calculator.