`(**) \ \ \ \ sin^n dx = -1/ncosxsin^(n - 1)x + (n - 1)/n int sin^(n-2) dx`
$\\$
`int_0^(pi/2) sin^5x dx`
`= (-1/5cosxsin^4x)|_0^(pi/2)+4/5int_0^(pi/2) sin^3xdx`
`=4/5int_0^(pi/2) sin^3xdx`
$=\left. {\left( {\dfrac{{ - 4}}{{15}}{\rm{\;}}\cos x \sin^2x} \right)} \right|_0^{\frac{\pi }{2}} + \dfrac{8}{{15}}$`\int_0^{\frac{\pi }{2}} \sin xdx `
`=8/15 int_0^(pi/2) sinx dx`
$=\left. {\left( {\dfrac{{ - 8\cos x}}{{15}}} \right)} \right|_0^{\frac{\pi }{2}}$
$\fbox{$=\dfrac{8}{15}$}$
