The nice thing about mathematics is that when you encounter something, preferably something useful, but it isn't "possible" yet, you just define it.
The integral of e(x²) (usually with a minus-sign and some constants) appears to be very useful in e.g. statistics (normal distribution), but it doesn't have a closed-form primitive function, i.e. its primitive function cannot be expressed in terms of a finite composition of what we call 'elementary functions' (polynomials, trig, log/exp, ...).
Here comes the mathematician: we just give it a new name, we define it as a new function (search for the error-function, Erf(x)).
The name of the math that deals with these problems is called "differencial algebra" (I know not a creative name).
There is a theorem by Louiville:
If are rational functions and is not a constant, then,
is elementary if and only if there exists a rational solution to the differencial equation,
On some open interval.
*)Hmmm... the Jordan Closed Curve theorem does not work here, interesting.