1. ## Triple Integral

Find a condition on $\displaystyle G$ meeting the requirement that the equation

$\displaystyle M = 4G \int \int \int_{\Omega} \frac{d^3p}{(2\pi)^3} \frac{M}{\sqrt{p^2 + M^2}}$

possesses a solution for $\displaystyle M > 0$, where the integration is carried out over the region $\displaystyle \Omega$, a sphere with radius $\displaystyle r$ in the $\displaystyle p$ space.

use the fact : $\displaystyle \int \sqrt{x^2 + a} ~dx = \frac{1}{2} \left[x\sqrt{x^2 + a} + a \ln \left(x + \sqrt{x^2 + a}\right) \right]$ for $\displaystyle a > 0$

I really don't have any idea to start doing this problem.
any comment would be appreciated.

2. Originally Posted by dedust
Find a condition on $\displaystyle G$ meeting the requirement that the equation

$\displaystyle M = 4G \int \int \int_{\Omega} \frac{d^3p}{(2\pi)^3} \frac{M}{\sqrt{p^2 + M^2}}$

possesses a solution for $\displaystyle M > 0$, where the integration is carried out over the region $\displaystyle \Omega$, a sphere with radius $\displaystyle r$ in the $\displaystyle p$ space.

use the fact : $\displaystyle \int \sqrt{x^2 + a} ~dx = \frac{1}{2} \left[x\sqrt{x^2 + a} + a \ln \left(x + \sqrt{x^2 + a}\right) \right]$ for $\displaystyle a > 0$

I really don't have any idea to start doing this problem.
any comment would be appreciated.

Obscure notation (at least from my experience) obviously $\displaystyle d^3p$ is meant to be a volume element but what is $\displaystyle p^2$?

CB