Question

Solve : $\int$ $si{n}^3$ $x$ $co{s}^2$ $x$ $dx$

Answer 1

$\int {\sin }^{3}x{\cos }^{2}xdx$

$=\int {\sin }^{2}x{\cos }^{2}x\sin xdx$

$=\int (1-{\cos }^{2}x)\left({\cos }^{2}x\right)\sin xdx$

$=\int ({\cos }^{2}x-{\cos }^{4}x)\sin xdx$

Let $\cos x=t$

$\Rightarrow \frac{dcosx}{dx}=\frac{dt}{dx}$

$\Rightarrow -\sin x=\frac{dt}{dx}$

$\Rightarrow \sin xdx=-dt$

$=\int (t^2-t^4)-dt$

$=\int t^{4}dt-\int t^{2}dt$

$=\frac{t^5}{5}-\frac{t^3}{5}+c$

$=\frac{{\cos }^{5}x}{5}-\frac{{\cos }^{3}x}{3}+c$

Hence, the answer is $\frac{{\cos }^{5}x}{5}-\frac{{\cos }^{3}x}{3}+c.$