Question Number 126571 by benjo_mathlover last updated on 22/Dec/20
$$\:\:\int\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\:\sqrt{\mathrm{tan}\:^{\mathrm{2}} {x}−\mathrm{1}}}\:{dx}\:?\: \\ $$
Answered by MJS_new last updated on 22/Dec/20
$$\int\frac{\sqrt{\mathrm{tan}\:{x}}}{\:\sqrt{\mathrm{tan}^{\mathrm{2}} \:{x}\:−\mathrm{1}}}{dx}=\int\sqrt{\frac{\mathrm{tan}\:{x}}{\mathrm{tan}^{\mathrm{2}} \:{x}\:−\mathrm{1}}}{dx}= \\ $$$$=−\frac{\mathrm{i}}{\:\sqrt{\mathrm{2}}}\int\sqrt{\mathrm{tan}\:\mathrm{2}{x}}{dx} \\ $$$$\mathrm{now}\:\mathrm{simply}\:\mathrm{put}\:{t}=\sqrt{\mathrm{tan}\:\mathrm{2}{x}}\:\mathrm{and}\:\mathrm{solve} \\ $$