How to compute the following integral using complex analysis? 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?

You can just use power series, since both $\sin$ and $\exp$ are entire functions.
$$ \sin(\exp(e^{i\theta})) = \sum_{n\geq 0}\frac{(-1)^n}{(2n+1)!}\exp((2n+1)e^{i\theta})=\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)!}\sum_{m\geq 0}\frac{(2n+1)^m e^{m i\theta}}{m!}\tag{1} $$
but $\int_{0}^{2\pi}e^{mi\theta}\,d\theta = 0$ unless $m=0$, so
$$ \int_{0}^{2\pi}\sin(\exp(e^{i\theta}))\,d\theta = 2\pi\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)!}=\color{red}{2\pi\sin(1)}. \tag{2}$$
