TITLE: Integration of $e^{-x^2}$
QUESTION [7 upvotes]: I know that this has been asked many times, but I am very interested in the indefinite integral of $e^{-x^2}$. It is stated and proved that there is not elementary way to solve this. So does this imply the fact that there is a way to solve this integral. Has this been done before? Since this is a continuous function, I presume that it must have a solution. Can someone direct me to where I can find the solution to this? I have looked everywhere and I get it in terms of the error function.

REPLY [3 votes]: As already said, the indefinite integral of $e^{-x^2}$ cannot be expressed on the form of a finite number of elementary functions. This is the case of the vast majority of the indefinite integrals. However, some of them are especially useful in maths or physics and one attributes a specific name to those which are frequently encountered, for example : Error function, Bessel functions, Gamma function and many others. 
The approach is similar to  the old case of the indefinite integral of the function $\frac{1}{x}$ to which the name of "logarithm" was attributed. The difference is that "logarithm" is so commonly used and is so usefull that it is now considered as an elementary function.
The non-elementary functions which are well-known, recorded and doccumented are referred as "special functions" : A discussion for general public can be found pp.18-36 in : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales