In mathematics, a nonelementary integral is an integral for which it can be shown that there exists no formula in terms of elementary functions (i.e. involving polynomials, and the standard functions sin, cos, exp, and so on). It can be shown (though not easily) that, if one is given a function of any complexity, the chances that it will have an elementary antiderivative are very low.
The evaluation of nonelementary integrals can often be done using Taylor series. This is because Taylor series can always be integrated as one would an ordinary polynomial, even if there is no elementary antiderivative of the function that generated the Taylor series.
However, in some cases it is not possible to rely on Taylor series. For example, if the function is not infinitely differentiable, one cannot generate a Taylor series. Even if a Taylor series can be generated, there a good possibility that it will diverge and not represent the function one is attempting to antidifferentiate. Many functions which are infinitely differentiable have higher order derivatives that are unmanageable by hand. In these cases, it is not possible to evaluate indefinite integrals, but definite integrals can be evaluated numerically, for instance by Simpson's rule.
The integrals for many of these functions can be written down if one allows so-called "special" (nonelementary) functions. For example, the first example's integral is expressible using incomplete elliptic integrals of the first kind, the second and third use the logarithmic integral, the fourth the exponential integral, and the fifth the error function. Still, there exist functions, such as xx and sin(sin(x)) for which no notation currently exists to describe their integrals (other than the use of the integrals themselves).
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