How could I prove
$$ \int_{0}^{1} \frac{\cos(xu)}{\sqrt{1 - u^2}} \, du = \int_{1}^{\infty} \frac{\sin(xu)}{\sqrt{u^2 - 1}} \, du \ $$
?
Do I need to use Cauchy's Residue Theorem or is it possible to prove without complex analysis?
P.S. Both should be equal to $\frac{\pi}{2} J_0(x)$, but I want to prove it.
