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?
2026-04-26 13:21:39.1777209699
How to determine the convergence/divergence of an improper integral $\int_{0}^{\infty} \frac{\arctan(ax) - \arctan(bx)}{x}dx$
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