integrate $\int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$ How to integrate
$ 1)\displaystyle \int_0^{2\pi} e^{\cos \theta} \cos( \sin \theta) d\theta$
$ 2)\displaystyle \int_0^{2\pi} e^{\cos \theta} \sin ( \sin \theta) d\theta$

Let $\gamma$ be the unitary circumference positively parametrized going around just once.
Consider $\displaystyle \int _\gamma \frac{e^z}{z}\,dz$.
On the one hand $$\begin{align}
\int _\gamma \frac{e^z}{z}\mathrm dz&=\int \limits_0^{2\pi}\frac{e^{e^{i\theta}}}{e^{i\theta}}ie^{i\theta}\mathrm d\theta\\
&=i\int _0^{2\pi}e^{\cos (\theta)+i\sin (\theta )}\mathrm d\theta\\
&=i\int _0^{2\pi}e^{\cos (\theta )}[\cos (\sin (\theta))+i\sin (\sin (\theta))\textbf{]}\mathrm d\theta.
\end{align}$$
On the other hand Cauchy's integral formula gives you: $\displaystyle \int _\gamma \frac{e^z}{z}\mathrm dz=2\pi i$.
$\large \color{red}{\text{FINISH HIM!}}$
