<s>[INST] How to integrate following integral How to integrate following integral i am having physics background and i am trying calculating the induced charge in a specific problem.
The Integral need to be evaluted is 
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{(x^2+y^2+b^2)^{3/2}}\,dxdy$$
 [/INST] Let $b\not=0$, then, by letting $x=r\cos(t)$ and $y=r\sin(t)$ (polar coordinates) the integral becomes
\begin{align*}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{(x^2+y^2+b^2)^{3/2}}\,dxdy&=
\int_{r=0}^{\infty}\int_{t=0}^{2\pi}\frac{1}{(r^2+b^2)^{3/2}}\,(rdrdt)\\
&=\left[-\frac{2\pi}{(r^2+b^2)^{1/2}}\right]_0^{\infty}=\frac{2\pi}{|b|}.
\end{align*}
P.S. The integral can be done also by using cartesian coordinates
\begin{align*}4\int_{0}^{\infty}\left(\int_{0}^{\infty}\frac{dx}{(x^2+y^2+b^2)^{3/2}}\right) dy
&=4\int_{0}^{\infty}\left[\frac{x}{(y^2+b^2)(x^2+y^2+b^2)^{1/2}}\right]_{x=0}^{\infty}dy\\
&=4\int_{0}^{\infty}\frac{dy}{y^2+b^2}
=\frac{4}{|b|}\left[\arctan\left(\frac{y}{|b|}\right)\right]_{y=0}^{\infty}\\&=\frac{2\pi}{|b|}
\end{align*}
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