3.56: Shaded area; tan AP and cos GP

The diagram shows part of the curve $y=\dfrac{4}{2x+1}$ and the straight line $2y=6x+1$. Find the area of the shaded region in the form $\mathrm{ln}\,a+b$, where $a$ is an integer and $b$ is a rational number.

The first three terms of an arithmetic progression are $\mathrm{tan}\,x$, $\dfrac{5}{2}\mathrm{tan}\,x$, $4\mathrm{tan}\,x$. Find the values of $x$ in $-180^{\circ}\leqslant x\leqslant 180^{\circ}$ for which the sum to $30$ terms is $455\sqrt{3}$.

The first three terms of a geometric progression are $\mathrm{sin}\,2\theta$, $\mathrm{sin}\,4\theta$, $\mathrm{sin}\,6\theta$, where $-\dfrac{\pi}{2}\leqslant\theta\leqslant\dfrac{\pi}{2}$ and $\mathrm{sin}\,2\theta\neq0$. Find the values of $\theta$ for which this progression has a sum to infinity.