Integral $\int_{-\pi}^\pi e^{\sin x} \sin 4x \ dx$

Question

I found this question in an old real analysis text book, (so old the cover had come off) I graphed it on wolfram alpha, and it looks like an almost odd function. I think the integral evaluates to zero. Would anyone care help prove? (assuming I'm right)

Answer 1

$\sin 4x = 2 \cos 2x \sin 2x = 4 (2 \sin^2 x -1 ) \sin x \cos x $

Substitute $t = \sin x$