Editing Closed-form expression (section)

=== Differential Galois theory ===
{{main|Differential Galois theory}}
{{See also|Nonelementary integral}}

The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as [[differential Galois theory]], by analogy with algebraic Galois theory.

The basic theorem of differential Galois theory is due to [[Joseph Liouville]] in the 1830s and 1840s and hence referred to as '''[[Liouville's theorem (differential algebra)|Liouville's theorem]]'''.

A standard example of an elementary function whose antiderivative does not have a closed-form expression is: <math display="block">e^{-x^2},</math> whose one antiderivative is ([[up to]] a multiplicative constant) the [[error function]]:
<math display="block">\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^x e^{-t^2} \, dt.</math>