Given that \(\int _ { 0 } ^ { \mathrm { a } } \left( 6 \mathrm { e } ^ { 2 \mathrm { x } } + \mathrm { x } \right) \mathrm { dx } = 42\), show that \(\mathrm { a } = \frac { 1 } { 2 } \ln \left( 15 - \frac { 1 } { 6 } \mathrm { a } ^ { 2 } \right)\).
Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 decimal places. Use a starting value of 1 and show the result of each iteration.