TITLE: Why do un-integrable functions exist?
QUESTION [12 upvotes]: By un-integrable I mean functions whose antiderivative can not be expressed in terms of elementary functions. 
I recently learnt that any differentiable function can be expanded using the Taylor expansion, which essentially provides a polynomial representation. And polynomial funcitons are very easy to integrate. Then, why is it that there are functions whose antiderivative can not be expressed in terms of elementary functions?

REPLY [7 votes]: Why do un-integrable functions exist ?

Look at it this way: The inverse of an elementary function is not always elementary, so why should its anti-derivative be such ? So, if you accept the fact that $f(x)=x+\sin x$ is elementary, but $f^{-1}(x)$ is not, you should also accept the fact that, even though $g(x)=e^{-x^2}$ is elementary, its antiderivative is not.