Express \(\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }\) in the form \(\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }\), where \(A\) and \(B\) are integers.
(3 marks)
Hence show that \(\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2\), where \(p\) is rational.
(4 marks)
Given that \(\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\) can be written as \(C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }\), find the value of \(C\).
(1 mark)
Hence find the exact value of the area of the region bounded by the curve \(y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = - 1\) and \(x = 0\).
You may assume that \(y > 0\) when \(- 1 \leqslant x \leqslant 0\).