4 The integral \(\mathrm { I } _ { \mathrm { n } }\) is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } \left( 1 + \mathrm { x } ^ { 5 } \right) ^ { \mathrm { n } } \mathrm { dx }\).
- By considering \(\frac { d } { d x } \left( x \left( 1 + x ^ { 5 } \right) ^ { n } \right)\), or otherwise, show that
$$( 5 n + 1 ) l _ { n } = 2 ^ { n } + 5 n l _ { n - 1 }$$
- Find the exact value of \(I _ { 3 }\).