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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 $\frac{\pm1-i}{2}$. –  bgins Apr 16 '12 at 10:30
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correction: complete the closed curve in the upper upper half plane, with residues at $i$ and $i\pm1$. –  bgins Apr 16 '12 at 12:52
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Now wait for the OP to try it before giving more hints or solutions! –  GEdgar Apr 16 '12 at 12:57
    
@GEdgar: thanks, good advice. It's certainly doable with a little work (partial fraction and residue calculations). –  bgins Apr 16 '12 at 15:39
    
@bgins Ample time has passed; perhaps you could write a detailed solution now? –  Lord_Farin May 22 '13 at 21:18

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