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How would I calculate the following integral? $$\int_{-\infty}^\infty \frac{1}{(x^2 + 1)(x^4+4)^2} dx$$ Part (a) says define Laurent's theorem for the Laurent series expansion and give the definition of a residue of a function at point $a$ which I've done, but I can't see how this would help in solving this question?

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    You need to complete the closed curve that is positively oriented. In this case, use $x=Re^{i\theta}$ for $\theta\in(-\pi,0)$ as $R\to\infty$ for the lower half-plane; the integral over this part of the path vanishes so you are left with the residues at $-i$ and $\frac{\pm1-i}{2}$.2012-04-16
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    *correction*: complete the closed curve in the upper upper half plane, with residues at $i$ and $i\pm1$.2012-04-16
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    Now wait for the OP to try it before giving more hints or solutions!2012-04-16
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    @GEdgar: thanks, good advice. It's certainly doable with a little work ([partial fraction](http://en.wikipedia.org/wiki/Partial_fractions_in_complex_analysis) and [residue](http://en.wikipedia.org/wiki/Residue_(complex_analysis)) calculations).2012-04-16
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    @bgins Ample time has passed; perhaps you could write a detailed solution now?2013-05-22

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