How to evaluate this integral?

Feb 2009
Hi! I have two integrals of the form:

\(\displaystyle \int_0^{2\pi}{d\theta \cos(\theta)e^{i[a \cos(\theta) + b \sin(\theta)]}}\)
\(\displaystyle \int_0^{2\pi}{d\theta \sin(\theta)e^{i[a \cos(\theta) + b \sin(\theta)]}}\)

I have tried to run this in Mathematica. Then it just gets stuck in an infinite loop. I guess it is because it tries to do integration by parts, which I also have tried without any sucess so far. So is there any way to evaluate the integral?
Mar 2010
Kinda reminds me of Euler's formula: \(\displaystyle e^{ix} = \cos x + i\sin x \)