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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.

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$\displaystyle\int x^{^{\tfrac x{\ln x}}}dx\qquad$ ;-) – Lucian Feb 18 '14 at 6:00
@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-) – David Feb 18 '14 at 6:04
It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.) – Martin Sleziak Jul 4 '14 at 8:04
up vote 11 down vote accepted

Try this link. A lot of simple functions, btw :)

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Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions". – David Feb 18 '14 at 6:38

Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.

However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea. (Liouville's theorem is part of what is called differential Galois theory)

If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.

You could also try Pete Goetz's presentation here which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.

Note: Proving that a certain function does not have an elementary antiderivative is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.

I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.

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