Question Number 123599 by aurpeyz last updated on 26/Nov/20
$$\int{tan}^{\mathrm{5}} {xdx} \\ $$
Answered by Dwaipayan Shikari last updated on 26/Nov/20
$$\int{tan}^{\mathrm{2}} {x}\:{tan}^{\mathrm{3}} {x}\:{dx} \\ $$$$=\int{tan}^{\mathrm{3}} {x}\:{sec}^{\mathrm{2}} {xdx}−\int{tan}^{\mathrm{3}} {x}\:{dx} \\ $$$$=\frac{{tan}^{\mathrm{4}} {x}}{\mathrm{4}}−\int{sec}^{\mathrm{2}} {x}\:{tanx}\:−\int{tanxdx} \\ $$$$=\frac{{tan}^{\mathrm{4}} {x}}{\mathrm{4}}−\frac{{tan}^{\mathrm{2}} {x}}{\mathrm{2}}+{log}\left({cosx}\right)+{C} \\ $$