Answer
$\dfrac {63\pi }{512}\approx 0.386$
Work Step by Step
If $n$ is even
$$\Rightarrow \int ^{\pi /2}_{0}\cos ^{n}xdx=\left( \dfrac {1}{2}\right) \left( \dfrac {3}{4}\right) \left( \dfrac {5}{6}\right) \ldots \left( \dfrac {n-1}{n}\right) \left( \dfrac {\pi }{2}\right) $$
$$\Rightarrow \int ^{\pi /2}_{0}\cos ^{10}xdx=\left( \dfrac {1}{2}\right) \left( \dfrac {3}{4}\right) \left( \dfrac {5}{6}\right) \left( \dfrac {7}{8}\right) \left( \dfrac {9}{10}\right) \left( \dfrac {\pi }{2}\right) =\dfrac {63\pi }{512}\approx 0.386$$
