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Question Number 167490 by mathlove last updated on 18/Mar/22

∫sin^3 xcos^2 xdx=?

$$\int{sin}^{\mathrm{3}} {xcos}^{\mathrm{2}} {xdx}=? \\ $$

Answered by nimnim last updated on 18/Mar/22

I=∫sin^2 cos^2 xsinxdx    =∫(1−cos^2 x)cos^2 xsinxdx    =∫cos^2 xsinxdx−∫cos^4 xsinxdx    =−(1/3)cos^3 x+(1/5)cos^5 x+C

$${I}=\int{sin}^{\mathrm{2}} {cos}^{\mathrm{2}} {xsinxdx} \\ $$$$\:\:=\int\left(\mathrm{1}−{cos}^{\mathrm{2}} {x}\right){cos}^{\mathrm{2}} {xsinxdx} \\ $$$$\:\:=\int{cos}^{\mathrm{2}} {xsinxdx}−\int{cos}^{\mathrm{4}} {xsinxdx} \\ $$$$\:\:=−\frac{\mathrm{1}}{\mathrm{3}}{cos}^{\mathrm{3}} {x}+\frac{\mathrm{1}}{\mathrm{5}}{cos}^{\mathrm{5}} {x}+{C} \\ $$

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