$0$ On the interval $[0,\pi]$

$\langle \cos\!\left(x\right),\sin^{3}\!\left(x\right) \rangle) =$

$\displaystyle\int$

$dx$

$\hskip 140pt u =$ $du =$

$\hskip 72pt$		$=$		$\displaystyle\int$				$du$
		$=$

$\Vert \cos\!\left(x\right)\Vert=\Bigg($

$\displaystyle\int$

$dx\Bigg)^\frac{1}{2} =$