$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $ [duplicate]

Study the convergence of the sequence$$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $$ and find its limit.

My idea:

$I_n= \int_{0}^{\frac{\pi}{2}} e^{-n \sin x}\,dx + \int_{\frac{\pi}{2}}^{\pi} e^{-n \sin x}\,dx $.Afterwards I did a change of variables in the second integral.

Let $y=\pi -x$. Then it becomes $\int_{0}^{\frac{\pi}{2}}e^{n \sin y}\,dy$.

Now I am stuck.