# Contour integral with pure imaginary pole of order two : $\int_{0}^{\infty}\frac{x^2}{(x^2+9)^2(x^2+4)}dx$

by dan   Last Updated July 12, 2019 06:20 AM

I am trying to calculate $$\int_{0}^{\infty}\frac{x^2}{(x^2+9)^2(x^2+4)}dx$$. I want to use the upper half-circle as the contour (since the integrand is even), yet the function as poles in $$2i , 3i$$, but the pole $$3i$$ is an order $$2$$ pole, I couldn't find a way to calculate the residue of $$3i$$ in order to calculate the integral. Is there a way to split the integrand to $$\frac{A}{x^2+9}$$ + $$\frac{B}{x^2+9}$$ + $$\frac{C}{x^2+4}$$? or simple way to calculate the residue to $$x=3i$$?

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