Show $ \int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2} $ if $ a < b$

Question

Show that if $a < b$:

$$ \int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2} $$

I could see solution by Fourier analysis or by Contour Integration, but why does the larger frequency $b$ not matter?