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I have a little problem with applying a Monte Carlo method: Importance Sampling. I need to calculate: $$ \int_0^\infty \int_0^\infty \frac{dx\;dy}{2 \pi (1 + x^2 + y^2)^{3/2}} $$ Can somebody help me ? Thanks in advance.

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What is the link between the title and the question? –  Did Dec 18 '11 at 20:04
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1 Answer

Since the integrand is radially symmetric, change to radial coordinates: $dx \;dy \rightarrow r \;dr \;d\theta$, and $x^2+y^2\rightarrow r^2$. After performing the $\theta$ integral (cancelling the $2\pi$), and making an additional $u=1+r^2$ substitution, you have $$ \int_0^\infty \frac{r \;dr}{(1+r^2)^{3/2}} = \frac{1}{2}\int_1^\infty u^{-3/2} \; du = -u^{-1/2}\bigg\vert_1^\infty = 1. $$

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