Question

Integrate: ${\sin }^{3}x{\cos }^{3}x$

Answer 1

$I=\int {\sin }^{3}x{\cos }^{3}xdx$

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

$I=\int {\sin }^{3}x(1-{\sin }^{2}x)\cdot \cos xdx$

Let $u=\sin x$

$du=\cos xdx$

$I=\int u^3(1-u^2)du$

$I=\int (u^3-u^5)dx$

$I=\frac{u^4}{4}-\frac{u^6}{6}+c$

$I=\frac{1}{4}{\sin }^{4}x-\frac{1}{6}{\sin }^{6}x+C$.