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.
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.