5.
$$4 x ^ { 2 } + 4 x + 17 \equiv ( 2 x + p ) ^ { 2 } + q$$
where \(p\) and \(q\) are integers.
- Determine the value of \(p\) and the value of \(q\)
Given that
$$\frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \equiv \frac { 1 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } + \frac { A x + B } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } }$$
where \(A\) and \(B\) are integers,
- write down the value of \(A\) and the value of \(B\)
- Hence use algebraic integration to show that
$$\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \mathrm {~d} x = k + \frac { 1 } { 2 } \ln k$$
where \(k\) is a rational number to be determined.