For what values of r does the integral convege?
\[\int\limits_{0}^{+\infty}\frac{ 1 }{ x^r(x+2) }dx\]
Well, i tried dividing the integral from 0 to 1 and from 1 to infinity and then compare with the integral of 1/x^r, but not sure how to compare the second one. Any ideas?

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So long as \(r\) is sufficiently large, it should converge.

Let's say that: \[
\frac{1}{x^{r+1}+2x^r} <\frac{1}{x^{r+1}}
\]and perhaps use a comparison test.

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