Let $\displaystyle J_0 (x) = 2/\pi \int_{0}^{\pi/2} cos (x cos \theta) d\theta$. Show that $\displaystyle \int_{0}^{\infty} J_0(x) \exp(-ax) dx = 1/\sqrt{(1+a^2)}$ if $\displaystyle a>0$.

I'm guessing that you have to reverse the order of integration in order to make the integrals easier to evaluate, but it still seems quite tricky. Any ideas?