# Evaluating a definite integral by changing variables.

How can I evalute this integral?

$$\psi(z)=\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty [(x-a)^2+(y-b)^2+z^2]^{-3\over 2}f(x,y)\,\,\,dxdy\;.$$

I think we can treat $z$ as a constant and take it out of the integral or something. Maybe changing variables like taking $u={1\over z}[(x-a)^2+(y-b)^2]$? But then how do I change $f(x,y)$ which is some arbitrary function, etc?

Thanks.

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I'm not saying I know how to compute this integral, but $\psi(x,y,z)$ doesn't make sense ; you should write $\psi(z)$ instead, since your double integral is just a function of $z$ ; the variables $x$ and $y$ in it could be replaced by any other symbol and the result would still be the same. (In french we have a term that says that the variables don't speak, they're... "mute"? I don't feel like this makes sense though. For those who speak French here, "les variables $x$ et $y$ sont muettes". If someone could translate this in mathematical terms it would be nice.) – Patrick Da Silva Dec 21 '11 at 9:31
@PatrickDaSilva: Thanks! You are right. My teacher call those variables "dummy variables". I have edited the question. – gareth Dec 21 '11 at 10:21

There's no general way to evaluate this integral, for if there were, you could integrate any function $g(x,y)$ by calculating this integral for $f(x,y)=g(x,y)[(x-a)^2+(y-b)^2+z^2]^{\frac32}$.
+1. Ergo.   – Did Dec 21 '11 at 11:21