5 A forest is burning so that, \(t\) hours after the start of the fire, the area burnt is \(A\) hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to \(A ^ { 2 }\).
- Write down a differential equation which models this situation.
- After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt.
- Show that the substitution \(u = \mathrm { e } ^ { x } + 1\) transforms \(\int \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x\) to \(\int \frac { u - 1 } { u } \mathrm {~d} u\).
- Hence show that \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x = \mathrm { e } - 1 - \ln \left( \frac { \mathrm { e } + 1 } { 2 } \right)\).