Evaluating $\int_{0}^{\infty}\sin(x)\sin\left(\frac{1}{x}\right)\,dx$

Question

Playing around with some integrals, I stumbled across a cute one: $$ \int_{0}^{\infty}\sin(x)\sin\left(\frac{1}{x}\right)\,dx $$ Substituting $z=\frac1x$ gives $$ \int_{0}^{\infty}\sin(x)\sin\left(\frac{1}x\right)\,dx = \int _0^{\infty} \frac{\sin(z)\sin\left(\frac{1}{z}\right)}{z^2}\,dz, $$which Mathematica gave as $\frac{\pi}{2}J_1(2)$, where $J_1$ is the Bessel function of the first kind. I'd be curious to see a derivation of this, as well as a rigorous proof the integral converges.