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How can I find closed-form expression of the following double integral $$ \int_{0}^{\pi/4}\int_{0}^{\infty}{{\rm d}r\,{\rm d}\phi \over u_{1}^{2} + u_{2}^{2}\,r + 2\,u_{1}u_{2}\,\sqrt{r\,}\,\cos\left(\phi\right)}\ {\large ?} $$ Please help me as soon as you can.

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Are you sure it is convergent? The integral with respect to $r$ diverges at infinity. –  Sasha Jun 29 '12 at 12:44
Are you missing a factor of like $1/r$ or $1/r^2$? –  kennytm Jun 29 '12 at 17:49
Hence the answer is: there is a closed-form expression, which is $+\infty$. –  Did Jul 1 '12 at 20:36

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