Editing Closed-form expression (section)

== Symbolic integration ==

[[Symbolic integration]] consists essentially of the search of closed forms for [[antiderivative]]s of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly [[logarithm]]s, [[exponential function]] and [[polynomial root]]s. Functions that have a closed form for these basic functions are called [[elementary function]]s and include [[trigonometric functions]], [[inverse trigonometric functions]], [[hyperbolic functions]], and [[inverse hyperbolic functions]].

The fundamental problem of symbolic integration is thus, given an elementary function specified by a closed-form expression, to decide whether its antiderivative is an elementary function, and, if it is, to find a closed-form expression for this antiderivative.

For [[rational function]]s; that is, for fractions of two [[polynomial function]]s; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved with [[partial fraction decomposition]]. The need for logarithms and polynomial roots is illustrated by the formula 

<math display="block">\int\frac{f(x)}{g(x)}\,dx=\sum_{\alpha \in \operatorname{Roots}(g(x))} \frac{f(\alpha)}{g'(\alpha)}\ln(x-\alpha),</math>

which is valid if <math>f</math> and <math>g</math> are [[coprime polynomials]] such that <math>g</math> is [[squarefree polynomial|square free]] and <math>\deg f <\deg g.</math>