Let $\alpha \in \mathbb R$ and C be the circle $\gamma(t)=e^\alpha t$, $-\pi\le t \le \pi$
Evaluate $$\int_{C}\frac{e^{\alpha z}}{z}dz.$$
Use the above, to show that $$\int_{0}^{\pi}e^{\alpha \cos t}\cos(\alpha \sin t)dt= \pi.$$
I want to use cauchy integral formula for this problem, but I do not know how to start. Can I use the circle $\gamma(t)=e^\alpha t$?