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Suppose we know that:

$$ \int \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \ dx = \log \left( \left| \arctan(x)\right| \frac{\sqrt {1 + x^2}}{\left|x\right|} + C \right) $$

Can we determine if the definite integral:$$ \int_0^\infty \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \ dx $$ converges, basing on the indefinite integral value?

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    $\begingroup$ Yes. Take limits as $x\to 0$ and $x\to \infty$ and analyze the results. $\endgroup$ – Mark Viola Jun 5 '15 at 14:02
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Yes, with some limitations.

Note, that it is not enough just to "plug in" the limit values. You must also check to see that the function is piecewise continous in that interval. In your case, this is true for all $x>0$, so it remains to be seen if the limits exists.

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  • $\begingroup$ do you mean: $\lim_{x\to\infty} \log \left( \left| \arctan(x)\right| \frac{\sqrt {1 + x^2}}{\left|x\right|} + c \right)$? $\endgroup$ – Elimination Jun 5 '15 at 14:29
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    $\begingroup$ @Elimination - yes, without the $C$.. $\endgroup$ – nbubis Jun 5 '15 at 14:30
  • $\begingroup$ Oh right. Thank you. $\endgroup$ – Elimination Jun 5 '15 at 14:31

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