How can changing the order of integration be used to solve the integral $\int_0^\infty \frac{\cos{(\alpha x)}-\cos{(\beta x)}}{x}\, dx$

Question

I am attempting to solve the integral $$\int_0^\infty \frac{\cos{(\alpha x)}-\cos{(\beta x)}}{x}\,dx$$ I know that the correct answer is $\ln{\frac{\beta}{\alpha}}$ and that the integrand can be rewritten as $$\int_\alpha^\beta \sin{(xy)}\,dy$$ to form the double integral $$\int_0^\infty \int_\alpha^\beta \sin{(xy)}\,dy\,dx$$ Now if we were integrating over a finite rectangular region, I would feel comfortable using Fubini's theorem to swap the order of integration. Is there a way to switch the order of integration for a double integral of this form?