Compute the following integral using complex analysis

Question

I'm trying to compute the following integral using complex analysis:
\begin{equation}
\int_0^{2\pi}\sin(\exp(e^{i \theta}))d\theta
\end{equation}
I know that there has to be an easy way out, but I can't see it.

I've tried the following: by changing of variable $z = e^{i\theta}$, we get to
\begin{equation}
\int_{|z|=1}\frac{\sin(\exp(z))}{iz}dz = \operatorname{Res}(f,0) = \lim_{|z|\to0}-i\sin(\exp(z)) = -i\sin(1)
\end{equation}
It doesn't seem right, though. Can anyone please help me out?

Answer 1

$$\int_{|z|=1} \frac{\sin(\exp(z))}{iz}\mathrm{d}z = 2\pi i\text{ Res}\left(\frac{\sin(\exp(z))}{iz}, 0\right) = 2\pi \lim_{z\to 0} \sin(\exp(z)) = 2\pi \sin(1)$$