Show that the area bounded by the curves $y=\frac{1}{x}$, $y=mx$ and $y=2mx$ is the same for every $m\geq 0$.
The book suggests to use polar coordinates. Therefore using $x=r \cos\theta$ $y=r \sin\theta$ we transform the curve $y=\frac{1}{x}$ to $r^2=\frac{1}{\sin\theta \cos\theta}$.
So basically we have to integrate the curve $r^2=\frac{1}{\sin\theta \cos\theta}$ between the angles $\tan^{-1}m$ and $\tan^{-1}2m$. Therefore the shaded area is $$\int\limits_{\tan^{-1}m}^{\tan^{-1}2m}\frac{1}{2\sin\theta \cos\theta} d\theta$$
Am I right so far? and most importantly how do I evaluate the above integral?