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How to determine the convergence/divergence of an improper integral: $$ \int_{0}^{\infty} \frac{\arctan(ax) - \arctan(bx)}{x}dx $$ I know that this integral can be written into the sum of two integrals by the additivity property: $$ \int_{0}^{1} \frac{\arctan(ax) - \arctan(bx)}{x}dx + \int_{1}^{\infty} \frac{\arctan(ax) - \arctan(bx)}{x}dx $$ In order for the original integral to converge, it is necessary that each of the two integrals obtained converges, if one of them diverges, then the original integral also diverges. I do not quite understand how to determine the divergence/convergence of the two integrals obtained, I am confused by 2 parameters (a and b), nothing effective comes to mind, how to simply consider the parameters a and b separately. Is there any more concise and effective determining to separating the convergence/divergence of these two integrals?

Little Mandelbrot
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  • Maybe this helps https://math.stackexchange.com/questions/1046074/show-that-int-0-infty-fracfax-fbxxdx-f0-l-ln-fracba – AAM Mar 09 '24 at 15:12
  • This is a Frullani integral. – Integrand Mar 09 '24 at 21:20
  • as usual, no one wants to think, really why think when a simple integral can be swatted by a big theorem. as a result, I figured out this question myself and dispensed with using Frulani's theorem, I wanted to publish a solution a little later without using Frulani's theorem, but the moderators - people who really have nothing to do in life except sit on this forum and earn a reputation - the only thing they can be proud of and brag about in their lives, closed this question – Little Mandelbrot Mar 14 '24 at 09:46

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