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?


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