Question

$\int si{n}^{2/3}xco{s}^{3}xdx$

Answer 1

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

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

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

$=\int {\sin }^{2/3}x\cos xdx-\int {\sin }^{2/3+2}x\cos xdx$

$=\int {\sin }^{2/3}x\cos xdx-\int {\sin }^{5/2}x\cos xdx$

Let $t=\sin x$

$\Rightarrow dt=\cos xdx$

$=\int t^{2/3}dt-\int t^{5/2}dt$

$=\frac{t^{2/3+1}}{2/3+1}-\frac{t^{5/2+1}}{5/2+1}+c$

$=\frac{t^{5/2}}{5/3}-\frac{t^{7/2}}{7/2}+c$ where c is the constant of integration

$=\frac{3(\sin x{)}^{5/3}}{5}-\frac{2(\sin x{)}^{7/2}}{7}+c$

$=\frac{3{\sin }^{5/3}x}{5}-\frac{2}{7}{\sin }^{7/2}x+c$ where $t=\sin x$.