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QUESTION Use the standard semicircular contour to show that
$$\int_{-\infty}^\infty\frac{dx}{(x^2+a^2)(x^2+b^2)}=\frac{\pi}{ab(a+b)},\
a,b>0$$

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ANSWER

$$I=\int_{-\infty}^\infty \frac{dx}{(x^2+a^2)(x^2+b^2)}=2 \pi i
(\textrm{Res}(ia)+\textrm{Res}(ib))$$

This has 4 poles, 2 inside the contour.\\
$\ds\textrm{Res}(z_0)=\frac{1}{2z_0(z_0^2+b^2)+2z_0(z_0^2+a^2)}\\
I=2 \pi
i\left(\frac{1}{2ia(b^2-a^2)}+\frac{1}{2ib(a^2-b^2)}\right)=\frac{\pi
(b-a)}{ab(b^2-a^2)}=\frac{\pi}{ab(a+b)}$


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