When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the examples I give mostly come from "folklore" or guesswork.
Can anyone point me to a list (not a complete list of course!) of fairly simple elementary functions whose antiderivatives are not elementary? I'm thinking of things like $\exp(x^2)$ which is the standard example, $\sin(\exp(-x))$ perhaps, things like this, not hugely complicated formulae.
Try this link. A lot of simple functions, btw :)
http://calculus-geometry.hubpages.com/hub/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
As was said in the comment below, the link doesn't work now.
Still, nothing could be deleted from the Internet permanently.
http://web.archive.org/web/20160612175604/http://hubpages.com:80/education/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives