I have trouble with the following problem:
Show that$$ \int_0^{\pi} e^{2\cos{\theta}}(\cos{(2\sin\theta})\sin\theta+\sin(2\sin\theta)\cos\theta)d\theta=\sinh2, $$ given the hint that it could be wise to study $\int_C e^zdz$ in several ways (with $ C: z(\theta)=2e^{-i\theta}$, with $0\leq\theta \leq \pi$, from a former problem).
The problem is that I can't really see why the hint will help, since I cannot get it equal to the integral in question. So could someone point me in some direction in which I could proceed?
Thanks in advance!
For the people who'll find this in about ten years or so, the integral is solved by expanding the integrand of the hint using Euler's formula and trigonometry. Then the integral in question should equal the real part, and you've got your solution using the definition of $ \sinh{2} $.