7 In this question you must show detailed reasoning.
- Find the values of \(A , B\) and \(C\) for which \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \equiv A + \frac { B x + C } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\).
- Hence express \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\) using partial fractions.
- Using your answer to part (b), determine \(\int _ { 0 } ^ { 2 } \frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \mathrm {~d} x\) expressing your answer in the form \(a + \ln b + c \pi\) where \(a\) is an integer, and \(b\) and \(c\) are both rational.