Question Number 96699 by Rio Michael last updated on 04/Jun/20
$$\int\:\frac{\mathrm{tan}^{\mathrm{3}} \left(\mathrm{ln}\:{x}\right)}{{x}}\:{dx}\:=\:?? \\ $$
Commented by bobhans last updated on 04/Jun/20
$$\mathrm{u}=\mathrm{ln}\left(\mathrm{x}\right)\:\Rightarrow\:\int\:\mathrm{tan}\:^{\mathrm{3}} \mathrm{u}\:\mathrm{du}\:=\:\int\left(\mathrm{sec}\:^{\mathrm{2}} \mathrm{u}−\mathrm{1}\right)\mathrm{tan}\:\mathrm{u}\:\mathrm{du} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}\:^{\mathrm{2}} \mathrm{u}+\:\mathrm{ln}\:\mid\mathrm{cos}\:\mathrm{u}\mid\:+\:\mathrm{c} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{ln}\mid\mathrm{x}\mid\right)\:+\:\mathrm{ln}\:\mid\mathrm{cos}\:\mathrm{ln}\mid\mathrm{x}\mid\mid\:+\:\mathrm{c} \\ $$
Commented by Rio Michael last updated on 04/Jun/20
$$\mathrm{perfect} \\ $$