7 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), where \(n\) is a non-negative integer. Show that
$$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$
Find the exact value of \(I _ { 4 }\).
7 Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x$, where $n$ is a non-negative integer. Show that
$$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$
Find the exact value of $I _ { 4 }$.
\hfill \mbox{\textit{CAIE FP1 2015 Q7}}