Question Number 20390 by tammi last updated on 26/Aug/17
$$\int\mathrm{sin}\:^{\mathrm{5}} {xdx} \\ $$
Answered by ajfour last updated on 26/Aug/17
$$\int\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}\right)^{\mathrm{2}} \mathrm{sin}\:{x}\:{dx} \\ $$$${let}\:\mathrm{cos}\:{x}={t}\:\:\Rightarrow\:\:\:−\mathrm{sin}\:{xdx}={dt} \\ $$$$\int\mathrm{sin}\:^{\mathrm{5}} {xdx}=−\int\left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\mathrm{2}} {dt} \\ $$$$\:\:=−\int\left(\mathrm{1}−\mathrm{2}{t}^{\mathrm{2}} +{t}^{\mathrm{4}} \right){dt} \\ $$$$\:\:=−{t}+\frac{\mathrm{2}{t}^{\mathrm{3}} }{\mathrm{3}}−\frac{{t}^{\mathrm{5}} }{\mathrm{5}}+{C} \\ $$$$\:\:\:=−\mathrm{cos}\:{x}+\frac{\mathrm{2}}{\mathrm{3}}\mathrm{cos}\:^{\mathrm{3}} {x}−\frac{\mathrm{1}}{\mathrm{5}}\mathrm{cos}\:^{\mathrm{5}} {x}+{C}\:. \\ $$