Evaluate \(\int_{0}^{\frac{1}{2} \pi} x \sin x \mathrm{~d} x\).

Given that \(I_{n}=\int_{0}^{\frac{1}{2} \pi} x^{n} \sin x \mathrm{~d} x\), prove that, for \(n\gt 1\),
\(I_{n}=n\left(\frac{1}{2} \pi\right)^{n-1}-n(n-1) I_{n-2}\)

By first using the substitution \(x=\cos ^{-1} u\), find the value of
\(\int_{0}^{1}\left(\cos ^{-1} u\right)^{3} \mathrm{~d} u\)
giving your answer in an exact form.