Thread: How to prove that a function has no closed-form integral?

We know from calculus books that some functions (such as e^(-x^2) and sinx/x) have no closed-form integrals but no proofs are given. I want to know where can I find proofs of such statements? Thanks.

such proofs usually use ideas well beyond the scope of calculus texts. for example:

http://modular.math.washington.edu/e...20b/Conrad.pdf

one common reference given for such proofs (and indeed, quoted in the link above) is: M. Rosenlicht, Integration in finite terms, American Math. Monthly 79 (1972), 963-972.

don't feel daunted if either of these is beyond what you have learned so far. it turns out that "integration in elementary terms" is not an elementary subject.

Thank you for the information.