7 Show that \(\frac { \mathrm { d } } { \mathrm { d } t } \left( t \left( 1 + t ^ { 3 } \right) ^ { n } \right) = ( 3 n + 1 ) \left( 1 + t ^ { 3 } \right) ^ { n } - 3 n \left( 1 + t ^ { 3 } \right) ^ { n - 1 }\). Let \(I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + t ^ { 3 } \right) ^ { n } \mathrm {~d} t\). Using the above result, or otherwise, show that $$( 3 n + 1 ) I _ { n } = 2 ^ { n } + 3 n I _ { n - 1 }$$ Hence evaluate \(I _ { 3 }\).