<s>[INST] $\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$ I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a complex variable $z= x + iy$:
$$\int_0^\infty \frac{x^2}{(x^2-4)(x^2-9)}\,\text dx$$
 [/INST] by using the partial fraction 
$$\int_{0}^{\infty }\frac{x^2}{(x^2-4)(x^2-9)}dx=(\int_{0}^{\infty }\frac{1}{5(x-2)}-\frac{1}{5(x-2)}+\frac{3}{10(x-3)}-\frac{3}{10(x+3)})dx$$
$$=\frac{1}{5}\log(\frac{x-2}{x+2})+\frac{3}{10}\log(\frac{x+3}{x-3})
$$
take the limit when $x=\infty$ to get
$$\lim_{x\rightarrow \infty }\log(\frac{x-2}{x+2})=0$$
$$\lim_{x\rightarrow \infty }\log(\frac{x+3}{x-3})=0$$
$$I=-\frac{1}{5}\log(-1)-\frac{3}{10}\log(-1)=-\frac{i\pi }{2}$$
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