4.
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \cos x \mathrm {~d} x , \quad n \geq 0$$
- Prove that \(I _ { n } = \left( \frac { \pi } { 2 } \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } , n \geq 2\).
- Find an exact expression for \(I _ { 6 }\).
[0pt]
[P5 June 2002 Qn 4]