An elementary function is any function (finitely) composed of addition, multiplication, exponentiation, and their inverses (subtraction, division, roots, and logs). All of the trig functions are elementary functions because they can be written in terms of exponents. Likewise, the inverse trig functions can be written in terms of logarithms. The derivative of every elementary function is an elementary function. However, the integral of an elementary function can be composed of elementary functions and special functions, or not representable in either way. Some examples of non-elementary functions are the Lambert W function, the gamma function, the error function, the exponential integral, the polylogarithms, and the sine integral. One of the most common methods of taking non-elementary integrals is by converting the function into an analytic series (such as a Taylor series) and integrating the resulting polynomial (this can restrict the domain of the function). Usually you'll attempt to integrate using other tricks like u-substitution and integration by parts, and if you encounter a non-elementary integral you'll usually put it in a standard form so that the resulting series or special function is in a familiar form. Hope this helps. It wasn't entirely clear what you were looking for or why.