Integral of $\cos(\cos x)$ over $[0,2\pi]$ How to compute the following integral?
$$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$
I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?

It obviously cannot be solved by computing the indefinite integral - this is a matter of special functions.
Is this a homework question?  What's the subject?  You obviously need to delve into the world of Bessel functions, for instance.
http://en.wikipedia.org/wiki/Bessel_function
One thing you can do is: perform suitable translations $u=t+ \alpha$ for $\alpha$ an adequate phase to have it fit into the above list.
With this you should be able to finish your exercise.  I don't know if there is a way to obtain a nicer result (it all depends on where you have seen the exercise).
