Problem

$$4x^2+4x+17\equiv(2x+p)^2+q$$

where $p$ and $q$ are integers.

(a) Determine the value of $p$ and the value of $q$

Given that

$$\frac{8x+5}{\sqrt{4x^2+4x+17}} \equiv \frac{1}{\sqrt{4x^2+4x+17}}+\frac{Ax+B}{\sqrt{4x^2+4x+17}}$$

where $A$ and $B$ are integers,

(b) write down the value of $A$ and the value of $B$

(c) Hence use algebraic integration to show that

$$\int_{1/3}^1 \frac{8x+5}{\sqrt{4x^2+4x+17}}\,dx = k+\frac12\ln k$$

where $k$ is a rational number to be determined.