# For which parameters the integrals are convergent?

For which constants $$\alpha$$, $$\beta$$, $$\gamma$$ the following two integrals are convergent: $$\iiint_{x^2+y^2+z^2\leq 1} \frac{dxdydz}{x^\alpha+y^\beta+z^\gamma}\\ \iiint_{x^2+y^2+z^2\geq 1} \frac{dxdydz}{x^\alpha+y^\beta+z^\gamma}$$

Attempt: I'm trying to use the so called Lebesgue layer integration method. It says that if $$(X,\Sigma,\mu)$$ is a measure space, $$f:X\to[0,\infty]$$ a measurable function and $$\phi: [0,\infty)\to[0,\infty)$$ is a monotone, continuously differentiable function such that $$\phi(0)=0$$ then $$\int_X (\phi\circ f) d\mu=\int_{0}^\infty\mu\{f>t\}dt.$$

So in our case it should be $$f=\frac{1}{x^\alpha+y^\beta+z^\gamma}$$, $$\phi(t)=t$$, and $$\mu$$ is the Lebesgue measure defined on the Borel sigma algebra on $$\mathbb{R^3}$$. The function $$f$$ is not non-negative but I think that can be fixed by writing it as a sum $$f=f_{+}-f_{-}$$. However it looks unreal to find $$\mu\{f>t\}$$ with such a complicated function. I tried to calculate it but failed. Maybe I'm just trying to solve the question in a wrong way. Any ideas?