7.
$$I _ { n } = \int \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
- Prove that for \(n \geqslant 2\)
$$I _ { n } = \frac { 1 } { n } \left( - \sin ^ { n - 1 } x \cos x + ( n - 1 ) I _ { n - 2 } \right)$$
Given that \(n\) is an odd number, \(n \geqslant 3\)
- show that
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x = \frac { ( n - 1 ) ( n - 3 ) \ldots 6.4 .2 } { n ( n - 2 ) ( n - 4 ) \ldots 7.5 .3 }$$
- Hence find \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 5 } x \cos ^ { 2 } x d x\)