Question Number 21965 by j.masanja06@gmail.com last updated on 07/Oct/17
$${integrate} \\ $$$$\int{sin}^{\mathrm{3}} {xdx} \\ $$
Answered by Tikufly last updated on 07/Oct/17
$$\:\:=\int\:\left(\mathrm{sin}^{\mathrm{2}} {x}\right)\mathrm{sin}{xdx} \\ $$$$\:\:=\int\left(\mathrm{1}−\mathrm{cos}^{\mathrm{2}} {x}\right)\mathrm{sin}{xdx} \\ $$$$\:\:\mathrm{Put}\:{u}=\mathrm{cos}{x}\rightarrow{du}=−\mathrm{sin}{xdx} \\ $$$$\:{I}=−\int\left(\mathrm{1}−{u}^{\mathrm{2}} \right){du}=−\int\mathrm{1}{du}+\int{u}^{\mathrm{2}} {du} \\ $$$$\:\:\:=−{u}+\frac{\mathrm{1}}{\mathrm{3}}{u}^{\mathrm{3}} +{C} \\ $$$$\:\:\:=−\mathrm{cos}{x}+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{cos}^{\mathrm{3}} {x}+{C} \\ $$