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Question Number 141534 by sarkor last updated on 20/May/21

Answered by som(math1967) last updated on 20/May/21

∫cos^2 xsin^8 xcosxdx  =∫(1−sin^2 x)sin^8 xd(sinx)  =∫sin^8 xd(sinx)−∫sin^(10) xd(sinx)  =(1/9)sin^9 x−(1/(11))sin^(11) x+C

$$\int\boldsymbol{{cos}}^{\mathrm{2}} \boldsymbol{{xsin}}^{\mathrm{8}} \boldsymbol{{xcosxdx}} \\ $$$$=\int\left(\mathrm{1}−\boldsymbol{{sin}}^{\mathrm{2}} \boldsymbol{{x}}\right)\boldsymbol{{sin}}^{\mathrm{8}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right) \\ $$$$=\int\boldsymbol{{sin}}^{\mathrm{8}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right)−\int\boldsymbol{{sin}}^{\mathrm{10}} \boldsymbol{{xd}}\left(\boldsymbol{{sinx}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{9}}\boldsymbol{{sin}}^{\mathrm{9}} \boldsymbol{{x}}−\frac{\mathrm{1}}{\mathrm{11}}\boldsymbol{{sin}}^{\mathrm{11}} \boldsymbol{{x}}+\boldsymbol{{C}} \\ $$

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