1. ## Definite integration proof

Prove that $\int_0^{2\pi}e^{\cos\theta}\cos(\sin\theta)=2\pi$

2. Differentiate under the integral sign, it'll be helpful to get what you want.

Define $f(\varphi ) = \int_0^{2\pi } {e^{\varphi \cos \theta } \cos (\varphi \sin \theta )} \,d\theta.$

From there, you just need to find $f(1),$ and it's done.

(You forgot the $d\theta.$)