Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the curve with equation

$$y=\sqrt{\frac{3x}{3x^2+5}}\qquad x\ge 0$$

The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the lines with equations $x=\sqrt{5}$ and $x=5$

The region $R$ is rotated through $360^\circ$ about the $x$-axis.

Use integration to find the exact volume of the solid generated. Give your answer in the form $a\ln b$, where $a$ is an irrational number and $b$ is a prime number.