Symbolic integration

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Symbolic integration is the application of computer software to solving integral problems in mathematics, but rather finding a symbolic expression instead of an exact numerical value.

For example:

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

is a symbolic result rather than a numerical value for the answer, where C is a constant of integration.

Finding the derivative of an expression is usually a straightforward process for which it is easy to determine an algorithm. The reverse question of finding the integral (of finding an expression whose derivative is the specified expression) is much more difficult. Many expressions that are relatively easy to state 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.

[edit] 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