8.
$$I _ { n } = \int _ { 0 } ^ { k } x ^ { n } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \quad n \geqslant 0$$
where \(k\) is a positive constant.
- Show that
$$I _ { n } = \frac { 2 k n } { 3 + 2 n } I _ { n - 1 } \quad n \geqslant 1$$
Given that
$$\int _ { 0 } ^ { k } x ^ { 2 } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 9 \sqrt { 3 } } { 280 }$$
- use the result in part (a) to determine the exact value of \(k\).