Question Number 141534 by sarkor last updated on 20/May/21
Answered by som(math1967) last updated on 20/May/21
$$\int\boldsymbol{{cos}}^{\mathrm{2}} \boldsymbol{{xsin}}^{\mathrm{8}} \boldsymbol{{xcosxdx}} \\ $$$$=\int\left(\mathrm{1}−\boldsymbol{{sin}}^{\mathrm{2}} \boldsymbol{{x}}\right)\boldsymbol{{sin}}^{\mathrm{8}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right) \\ $$$$=\int\boldsymbol{{sin}}^{\mathrm{8}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right)−\int\boldsymbol{{sin}}^{\mathrm{10}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{9}}\boldsymbol{{sin}}^{\mathrm{9}} \boldsymbol{{x}}−\frac{\mathrm{1}}{\mathrm{11}}\boldsymbol{{sin}}^{\mathrm{11}} \boldsymbol{{x}}+\boldsymbol{{C}} \\ $$