Question

Evaluate

${\int }_0^{\frac{\pi }{2}}(cos{)}^{5}xdx$

Answer 1

We have,

${\int }_0^{\frac{\pi }{2}}{\cos }^{5}xdx$

Let

$I_5={\int }_0^{\frac{\pi }{2}}{\cos }^{5}xdx$

Now,

${\int }_0^{\frac{\pi }{2}}{\cos }^{5}xdx=I_5$

$=\frac{4}{5}{I}_3$

$=\frac{4}{5}\times \frac{2}{3}\times I_1$

$=\frac{8}{15}{\int }_0^{\frac{\pi }{2}}\cos xdx$

$=\frac{8}{15}{{\left[\sin x\right]}_0}^{\frac{\pi }{2}}$

$=\frac{8}{15}[\sin \frac{\pi }{2}-\sin 0^o]$

$=\frac{8}{15}\times 1$

$=\frac{8}{15}$

Hence, this is the answer.