4.
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } x ^ { n } \sin 2 x \mathrm {~d} x , \quad n \geqslant 0$$
- Prove that, for \(n \geqslant 2\),
$$I _ { n } = \frac { 1 } { 4 } n \left( \frac { \pi } { 4 } \right) ^ { n - 1 } - \frac { 1 } { 4 } n ( n - 1 ) I _ { n - 2 }$$
- Find the exact value of \(I _ { 2 }\)
- Show that \(I _ { 4 } = \frac { 1 } { 64 } \left( \pi ^ { 3 } - 24 \pi + 48 \right)\)