Determine if the integral converges: $\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$

Question

Determine if the integral converges:

$$\int_1^{\infty} \frac{\arctan (px)}{x^q}dx$$

where $p,q\in\Bbb R$.

Answer 1

The convergence is determined by whether $q>1$. If so, then yes; if not, then no. The parameter $p$ plays no part in this determination, except (as @coco mentions below) where, when $p=0$, the integral always converges.

Answer 2

If $p\neq 0$ $$\frac{\arctan(px)}{x^q}\sim_\infty\frac{\pm\pi}{2x^q}$$ according to the sign of $p$ so the integral is convergent if $q>1$

and if $p=0$ the integral is convergent for all $q\in\mathbb{R}$.