TITLE: Evaluate $\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt$
QUESTION [6 upvotes]: What is the value of
$$\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt?$$

REPLY [4 votes]: The integral may be expressed at
$$\frac12 \operatorname{Re} \int_0^{2 \pi} dt \, e^{a e^{i t}} $$
The integral may be written in complex form as, letting $z=e^{i t}$:
$$-i \oint_{|z|=1} dz \frac{e^{a z}}{z} $$
which by Cauchy's integral theorem is simply $2 \pi$.  Thus, the sought-after integral is $\pi$.