I've wondered this, too! It helps that elementary functions are in some sense dense in the set of all other functions. Results like Stone-Weierstrass show that

I've wondered this, too! It helps that elementary functions are in some sense dense in the set of all other functions. Results like Stone-Weierstrass show that on a compact interval, any function can be arbitrarily-well approximated by a polynomial. Other things that might interest you-- how one might know if a function even has an antiderivative that can be expressed as an elementary function: http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differen...