Question

Find the integrals of the functions ${\sin }^{3}x{\cos }^{3}x$

Answer 1

Let $I=\int {\sin }^{3}x{\cos }^{3}x\cdot dx$
$=\int {\cos }^{3}x\cdot {\sin }^{2}x\cdot \sin x\cdot dx$
$=\int {\cos }^{3}x(1-{\cos }^{2}x)\sin x\cdot dx$
Put $\cos x=t$
$\Rightarrow -\sin xdx=dt$
$\Rightarrow I=-\int t^3(1-t^2)dt$
$=-\int (t^3-t^5)dt$
$=-\{\frac{t^4}{4}-\frac{t^6}{6}\}+C$
$=-\{\frac{{\cos }^{4}x}{4}-\frac{{\cos }^{6}x}{6}\}+C$
$=\frac{{\cos }^{6}x}{6}-\frac{{\cos }^{4}x}{4}+C$