Nonelementary integral

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.

Examples

Examples of functions with nonelementary antiderivatives include:

<math>\sqrt{1 - x^4}</math> (logarithmic integral)
<math>e^{-x^2}</math>

The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.

Nonelementary integral

Examples

See also

References

Further reading