TITLE: Evaluate $\int_0^\infty \frac{\arctan(3x) - \arctan(9x)}{x} {dx}$
QUESTION [10 upvotes]: Evaluate the integral $$\int_0^\infty \frac{\arctan(3x) - \arctan(9x)}{x} {dx}.$$

I tried to split this into 2 integrals and then using the substitution $t = \arctan(3x)$ but I got nowhere.

REPLY [10 votes]: An easy way to evaluate the integral is using Frullani's theorem
$$\int_0^\infty \frac{f(ax)-f(bx)}{x}\,dx=\bigg[f(0)-f(\infty)\bigg]\ln\left(\frac{b}{a}\right)$$
Taking $f(x)=\arctan(x)$ then the integral is simply evaluated to
$$-\frac{\pi}{2}\ln3$$