As part of an exercise for quantum mechanics I have to solve the following integral:
$$ \int \delta\left( a- \frac{x^2}{b} \right)\; \mathrm{d}^nx = a^{\frac{n}{2}-1} \cdot b^{\frac{n}{2}}$$
I do have the solution to the integral, but I can't figure out how to get to it. I know I have to use the following identity:
$$\delta\left( x^2 - a^2 \right) = \frac{1}{2|a|}\left[ \delta \left(x-a \right) + \delta \left( x+a \right) \right] $$
It would be great if someone could give me some hints.
edit 1:
By using the identity from above I get: $$\delta\left(a-\frac{x^2}{b}\right) = \frac{1}{2 \sqrt{a}} \left[\delta\left(\sqrt{a}-\frac{x}{\sqrt{b}}\right) + \delta\left(\sqrt{a}+\frac{x}{\sqrt{b}}\right)\right]$$
Use geometry and recognize the surface of a known object. You can find the answer of your question in the following images.
https://i.stack.imgur.com/eBcrZ.jpg https://i.stack.imgur.com/ROv6S.jpg