I wanted to calculate:
$$\int\limits_{|z|=4} \frac{\sin(e^z)}{z} dz$$
Using the Cauchy integral form, it's easy to check that:
$$=\int\limits_{|z|=4} \frac{\sin e^z}{z-(0)} = 2\pi i \sin(e^0) = 2\pi i \sin(1) $$
But... then I thought, what if I compute the contour integral directly, without the use of Cauchy's theorem?
Evaluating the parametrization $z=4e^{i\theta}$:
$$ \begin{split} &=\int\limits_0^{2\pi} \frac{\sin\big(e^{4e^{i\theta}}\big)}{4e^{i\theta}} 4ie^{i\theta}d\theta\\ &=i\int\limits_0^{2\pi} \sin\big(e^{4e^{i\theta}}\big) d\theta \end{split} $$
and from there I'm stuck.
Any hint on the integral will be appreciated <3.