how to evaluate this double integral by change the order?

Question

$\int_{0}^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$
I represented the integrand by an integral of $\sin(xy)$ to get an double integral $\int_{0}^{\infty}\int_{b}^{a} \sin(xy)dydx$
Then I want to change the order of this double integral, but I how can I solve $\int_{0}^{\infty}\sin(xy)dx$ first?

Answer 1

Your last integral does not exist, so you cannot use this method. For solving it, have a look at Proof of Frullani's theorem