I found the integral $$ \int_C e^{-1/z}\sin(1/z)dz $$ over the circle $|z|=1$ while doing some problems in Schaum's Outline for Complex Variables.
This integral has me stumped. The answer is $2\pi i$, but I can't see why. There are no poles, so I don't think I can apply the residue theorem. Reparametrizing with $z=e^{it}$ for $t\in(0,2\pi)$ looks very messy.
How can this integral be approached? I tried to find the Taylor series, and the first few terms are $$ -\frac{1}{2z^2}-\frac{2}{3!z^3}-\cdots $$ so $0$ looks like an essential singularity, but I don't know if that's useful.