Question Number 223193 by Tawa11 last updated on 17/Jul/25
$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{tan}^{−\:\mathrm{1}} \left(\mathrm{r}\:\mathrm{sin}\:\theta\right)\:\mathrm{d}\theta\:\:\:=\:\:\:\mathrm{2}\boldsymbol{\chi}_{\mathrm{2}} \left(\frac{\sqrt{\mathrm{1}\:\:+\:\:\mathrm{r}^{\mathrm{2}} }\:\:−\:\:\mathrm{1}}{\mathrm{r}}\right) \\ $$