How to evaluate $\int_0^{2\pi} e^{e^{i\theta}}{d\theta}$?

Question

I have tried using a parametrization with $z = e^{i\theta}$ to no avail. I would greatly appreciate any solutions or suggestions please. Thanks

Answer 1

With $z=e^{i\theta}, dz=izd\theta$, so

$\int_0^{2\pi} e^{e^{i\theta}}{d\theta}=\frac{1}{i}\int_{|z|=1} \frac{e^z}{z}{dz}=2\pi$ by the residue theorem