Answer
$\dfrac {128}{315}\approx 0.406$
Work Step by Step
If $n$ is odd $(n\geq3)$
$$\Rightarrow \int ^{\pi /2}_{0}\cos ^{n}xdx=\left( \dfrac {2}{3}\right) \left( \dfrac {4}{5}\right) \left( \dfrac {6}{7}\right) \ldots \left( \dfrac {n-1}{n}\right) $$
$$\Rightarrow \int ^{\pi /2}_{0}\cos ^{9}xdx=\left( \dfrac {2}{3}\right) \left( \dfrac {4}{5}\right) \left( \dfrac {6}{7}\right) \left( \dfrac {8}{9}\right) =\dfrac {128}{315}\approx 0.406$$
