Symbolic integration

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

:$frac\{dF\}\{dx\}\; =\; f(x).$

This is also denoted

:$F(x)\; =\; int\; f(x)dx.$

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 nowadays 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.

Example

For example:

:$int\; x^2,dx\; =\; frac\{x^3\}\{3\}\; +\; C$

is a symbolic result for an indefinite integral (here C is a constant of integration), whereas

:$int\_\{-1\}^1\; x^2,dx\; =\; frac\{2\}\{3\}$

is a numerical result for a definite integral.

ee also

* Antiderivative
* Elementary function
* Risch algorithm

References

*Symbolic Integration 1 (transcendental functions) by Manuel Bronstein, 1997 by Springer-Verlag, ISBN 3-540-60521-5
*Joel Moses, Symbolic integration: the stormy decade, Proceedings of the second ACM symposium on Symbolic and algebraic manipulation, p.427-440, March 23-25, 1971, Los Angeles, California, United States

External links

*MathWorld|urlname=RischAlgorithm|title=Risch Algorithm|author=Bhatt, Bhuvanesh
* [http://integrals.wolfram.com Free online symbolic integrator]