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?
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1https://en.wikipedia.org/wiki/Frullani_integral – Angina Seng Oct 18 '19 at 03:18
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You could replace $\infty$ by a large real number $N$ and then let $N\to\infty$. – Angina Seng Oct 18 '19 at 03:19
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2Possible duplicate of Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $ – metamorphy Oct 18 '19 at 04:49