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?

Your integral follows from the identity of the Bessel function

$$ J_n (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(n \tau - x \sin(\tau))} \,d\tau  $$

with $n=0$.
