Nonelementary integral

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative 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 which elementary functions have elementary antiderivatives.

Examples

Examples of functions with nonelementary antiderivatives include:

; for the antiderivative can be written in terms of the exponential integral; for in terms of the error function; for any positive integer, the antiderivative elementary.

Some common non-elementary antiderivative functions are given names, defining so-called special functions, and formulas involving these new functions can express a larger class of non-elementary antiderivatives. The examples above name the corresponding special functions in parentheses.

Properties

Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence. However, even if the integrand has a convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral Taylor series.
Even if it isn't always possible to evaluate the antiderivative in elementary terms, one can approximate a corresponding definite integral by numerical integration. There are also cases where there is no elementary antiderivative, but specific definite integrals can be evaluated in elementary terms: most famously the Gaussian integral
The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.