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Determine if the integral converges:

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

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

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2 Answers 2

up vote 2 down vote accepted

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.

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@Ron Gordon I tried doing intergration by part, And I've got $\int_1^{\infty} \frac{x^{-q+1}}{p^2x^2+1}dx$ isn't $x^2$ affects the range? – StationaryTraveller Apr 12 '13 at 13:10
If $p=0$ then the integral converges for all $q$ – Cocopuffs Apr 12 '13 at 13:12
I do not understand the motivation for integrating by parts. The question of convergence is a matter of comparison. Arctan is constant at infinity, so the integral only converges when it decays faster than $1/x$. The behavior at $x=1$ does not affect the question. – Ron Gordon Apr 12 '13 at 13:12
@Cocopuffs: true, true. Thanks. – Ron Gordon Apr 12 '13 at 13:13
@TheAlchemist: correct. – Ron Gordon Apr 12 '13 at 13:44

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 $q\in\mathbb{R}$.

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Thanks mate. nice answer. – StationaryTraveller Apr 12 '13 at 13:25
You're welcome. – user63181 Apr 12 '13 at 13:29

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