Finding $\int_0^\infty \frac{\arctan(p\cdot x)\cdot \arctan(q\cdot x)}{x^2} \text{d}t$

Question

I am attempting to derive the value of the integral $$ I(p,q)= \int\limits_0^\infty \frac{\arctan(p\cdot x)\cdot \arctan(q\cdot x)}{x^2} \text{d}x $$ Differentiating the I w.r.t. p and then q gives the expression $$ \frac{\partial^2 I}{\partial p \, \partial q} = \frac{\pi}{2(p+q)} $$

Now I want to solve this equation but unclear as to how the constant(s) of integration may be found.

Answer 1

Use the fact that $I(p,0) = I(0,q) = 0$.