Problem

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

Solutions relying on calculator technology are not acceptable.

Figure 2 shows a sketch of the curve with equation

$$y=\frac{16\sin 2x}{(3+4\sin x)^2}\qquad 0\leq x\leq \frac{\pi}{2}$$

The region $R$, shown shaded in Figure 2, is bounded by the curve, the $x$-axis and the line with equation $x=\dfrac{\pi}{6}$

Using the substitution $u=3+4\sin x$, show that the area of $R$ can be written in the form $a+\ln b$, where $a$ and $b$ are rational constants to be found.