Question

Solve the integral:-
$\int {\cos }^{3}x{\sin }^{5}xdx=?$

Answer 1

$\int {\cos }^{3}x{\sin }^{5}xdx$

$=\int \cos x{\cos }^{2}x{\sin }^{5}xdx$

$=\int \cos x(1-{\sin }^{2}x){\sin }^{5}xdx$

$=\int ({\sin }^{5}x-{\sin }^{7}x)\cos xdx$

Let $t=\sin x\Rightarrow dt=\cos xdx$

$=\int (t^5-t^7)dt$

$=\frac{t^6}{6}-\frac{t^8}{8}+c$

$=\frac{{\sin }^{6}x}{6}-\frac{{\sin }^{8}x}{8}+c$