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I have an integral

$$ \int_{-\infty}^{\infty}\frac{1}{(\omega^{2}-4)(\omega-2-i)(\omega+2-i)}d\omega $$

And I wish to evaluate this using Cauchy's Integral Theorem. I understand that with a simple pole on the real axis like

$$ \frac{sin(x)}{x} $$

We can break the contour around $x=0$ and use Jordan's Lemma as the real axis goes to infinity. However I'm still unconfident in dealing with two poles on the real axis ($\omega=\pm2)$. How should I go about this?

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Hint: All the poles are simple, so you could break the integrand into a sum of four simple fractions of the form $\frac{c_k}{w-p_k}$, right?

Then just deal with each integral separately.

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