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 formula for 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 is sought at a particular input or set of inputs, rather than a general formula for F. 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.
[3] Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving many definite integrals encountered by practical engineering applications.
The main problem in the classical approach of symbolic integration is that, if a function is represented in closed form, then, in general, its antiderivative has not a similar representation.
This recurrence relation allows a fast computation of the Taylor series, and thus of the value of the function at any point, with an arbitrary small certified error.
This makes algorithmic most operations of calculus, when restricted to holonomic functions, represented by their differential equation and initial conditions.