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i found a sinle pole with residue [itex]\frac{\log{2i}}{2i}[/itex] at [itex]z=i[/itex] and a double pole with residue [itex]-\frac{1}{2i}-\frac{\log{2i}}{2i}[/itex] at [itex]z=-i[/itex]. this seems fair enough but i can't decide where to put my contour so as to enclose them both. the reside theorem tells me they will integrate to [itex]-\pi[/itex] if i can get them both inside the contour which is probably a good aim consdiering the next part of the question is to show

[itex]\int_0^{\infty} \frac{\log{x^2+1}}{x^2+1}dx=\pi \log{2}[/itex]

so

(i) where do i put the contour

(ii) how can i evaluate the next part given that my complex integral is in terms of z's and the one i want is in terms of x^2's so it isn't as simple as taking the real part after i use Jordan's lemma on teh semi circle is it?

thanks