Question Number 134693 by Engr_Jidda last updated on 06/Mar/21
$$\int{sec}^{\mathrm{4}} {xtanxdx}?? \\ $$
Answered by john_santu last updated on 06/Mar/21
$$\mathcal{X}=\int\left(\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{1}\right)\mathrm{tan}\:{x}\:\mathrm{sec}\:^{\mathrm{2}} {x}\:{dx} \\ $$$${let}\:\mathrm{tan}\:{x}\:=\:\rho\: \\ $$$$\mathcal{X}=\int\left(\rho^{\mathrm{2}} +\mathrm{1}\right)\rho\:{d}\rho\:=\:\frac{\mathrm{1}}{\mathrm{4}}\rho^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{2}}\rho^{\mathrm{2}} \:+\:{c} \\ $$$$\mathcal{X}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\mathrm{tan}\:^{\mathrm{2}} {x}\left(\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{2}\right)\:+\:{c}\: \\ $$