Symbolic integration

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 a 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 the integral of an elementary function (function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four 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 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 by Manuel Bronstein.

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