(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The integral I'm trying to solve is sqrt(x)/(1+x^2) from 0 to infinity.

2. Relevant equations

3. The attempt at a solutionI've attached my solution. I know it's not right, as I shouldn't get an imaginary solution. The answer is actually pi/sqrt(2) according to my book. The author used a keyhole contour (branch cut along positive real axis and avoiding the branch point at z=0), but I don't see what's wrong with my approach. Any help would be greatly appreciated.

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# Contour integral w/ sqrt

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