Prove the following equality: $\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi$ Prove the following equality: 
$$
\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi
$$

Hint :
Consider the other integral
$S=\int_0^\pi e^{4cos(t)}sin(4sin(t))dt$
and the sum
$C+iS=\int_0^\pi e^{4(cos(t)+isin(t))}dt$
$=\int_0^\pi e^{4e^{it}}dt$
put $4e^{it}=u$ or $t=-iln(u)+i2ln(2)$
the integral becomes
$i\int_{-1}^{1}\frac{e^udu}{u}$
I'll surely be downvoted.
