I was given a problem that asks me to approximate the integral $$ \int_0^\pi e^{i(t \sin x - n x)}dx $$ for large positive real values of $t$.
Are there any approximation methods that might be applicable? I have only learned the saddle point approximation, but it does not seem to work here, since the integrand is a unit complex number that oscillates in its direction very quickly. I truly have no clue as to how to even start approaching this.
Since this is a homework problem (or at least part of it), subtle hints are more welcomed than solutions. I'd like to solve it myself and then post it as an answer to this question if possible.
Thank you very much.