Let $r > 0$ then how to compute an integral $\int_{\partial B(0,r)}|x|^{-n}dS(x)$ where $S(x)$ is a standard spherical measure. I tried to move to polar coordninates, but i have difficulties with the $S(x)$. How i can move from $dS(x)$ to regural $dx$?
I was able to compute the integral $\int_{B(0,r)}|x|^{-n}dx$ using the fact that $|x|$ is a spherical function. Can i apply this result somehow with my problem?
In your special case, the integrand $|x|^{-n}$ has the constant value $r^{-n}$ on the sphere $\partial B(0,r)$, so $$\int_{\partial B(0,r)} |x|^{-n} dS(x) = \int_{\partial B(0,r)} r^{-n} dS(x) = r^{-n} \int_{\partial B(0,r)} dS(x) = r^{-n} \operatorname{Area}\bigl( \partial B(0,r) \bigr) . $$ (If you're in $\mathbf{R}^m$, the area is $r^{m-1}$ times the area of the unit sphere in $\mathbf{R}^m$. For $m=3$, it's the familiar $4 \pi r^2$.)