8 A surge in the current, \(I\) units, through an electrical component at a time, \(t\) seconds, is to be modelled. The surge starts when \(t = 0\) and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, \(I\) is modelled by the following differential equation.
\(\left( 2 t - t ^ { 2 } \right) \frac { d l } { d t } = \left( 2 t - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - 2 ( t - 1 ) l\)

1. By using an integrating factor show that, according to the model, while the surge is occurring, \(I\) is given by \(\mathrm { I } = \left( 2 \mathrm { t } - \mathrm { t } ^ { 2 } \right) \left( \sin ^ { - 1 } ( \mathrm { t } - 1 ) + 5 \right)\). The surge lasts until there is again no current through the component.
2. Determine the length of time that the surge lasts according to the model.
3. Determine, according to the model, the rate of increase of the current at the start of the surge. Give your answer in an exact form.