I would like to ask why, or how the expression
$$J{_n}(x)=\frac{i^{-n}}{2\pi}\int_{\frac{-3\pi}{2}}^{\frac{\pi}{2}}e^{i(x cos \phi+n\phi)}d\phi $$
is the same or leads to the following:
$$J{_n}(x)=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{i(x cos \phi)}cos{(n\phi)}d\phi $$
I have tried substitutions and all sorts of simplifications with the symmetry of cosine and sine. Please help me and include all the possible details.
Thank you very much!