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I have the following integral:

$\displaystyle\int\limits_{0}^\infty\frac{\arctan(\eta_2+x)}{(x+\eta_1)^2+\eta_3^2}\ d x$

where $\eta_1$, $\eta_2$ and $\eta_3$ are real numbers. If the integrand were even it could be evaluated through the residue theorem because limits of the integral could be changed to form $-\infty$ to $\infty$ and a semicircular contour in the upper (or lower) half plane could be used. But in this case I can't figure out how to close the region to use the residue theorem. I can't also find another method to evaluate the integral.

Can someone please offer a solution to the conclusion of this problem?

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  • 2
    $\begingroup$ i would start by taking a deruivative w.r.t $\eta_2$ $\endgroup$ – tired Jul 19 '16 at 12:51
  • $\begingroup$ ....and the final integral will look distusting, regardless what you are doing. $\endgroup$ – tired Jul 19 '16 at 13:09
  • $\begingroup$ What are the signs of $\eta_i$'s? We should know their signs if we want to do contour integration (otherwise, we should do case work, which is very annoying) $\endgroup$ – user258700 Jul 19 '16 at 13:11
  • $\begingroup$ Here $η_1$ is negative and $η_3$ is positive. $η_2$ could be either positive or negative. So we should do case work for only η2. $\endgroup$ – Banx Jul 19 '16 at 13:49
  • $\begingroup$ $$\int_{0}^{+\infty}\frac{\arctan(x+1)}{x^2+1}\,dx $$ does not have a nice closed form, so to crack the general integral with three parameters is quite hopeless, IMHO. What is the purpose of the computation of such integral? $\endgroup$ – Jack D'Aurizio Jul 19 '16 at 18:42

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