Integral in n-dimensional spherical coordinates

Not a very clever approach, but we can compute this integral using hyper-spherical coordinates. We use the coordinate system $r, \phi_1,...,\phi_{n-1}$ with $r > 0, \phi_{n-1} \in [0,2\pi)$ and $\phi_i \in [0, \pi]$ for all other $i$.

Then $$ \int_{B_1} \frac{dx}{\|x \|^m} = \int_{0}^{1} \int_{0}^{\pi} \cdots \int_{0}^{2\pi} \left( \frac{1}{r^m} \right) (r^{n-1} \sin^{n-2}(\phi_1) \sin^{n-3}(\phi_2) \cdots \sin(\phi_{n-1})) \, dr\, d\phi_1 \, \cdots d \phi_{n-1}. $$ The second term is the Jacobian coming from the coordinate change. We see that this integral is almost just the integral of a volume of a ball. We transform coordinates so that it is in fact exactly this. If $n-1-m \neq -1$ we let $s = r^{n-m}/(n-m)$ so $ds = r^{n-m-1} dr$. If $n-1-m = -1$, then we let $s = \log r$.

Then in this coordinate system the integral is (when $n-m-1 \neq -1$)

$$ \int_{0}^{1/(n-m)} \sin^{n-2}(\phi_1) \sin^{n-3}(\phi_2) \cdots \sin(\phi_{n-1})) \, ds\, d\phi_1 \, \cdots d \phi_{n-1}. $$ Which is precisely the volume of $B_{1/(n-m)}.$ This volume is known to be $$ \frac{\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}+1\right)}\left(\frac{1}{n-m}\right)^n. $$

Perhaps a more enlightened approach to this problem would be to realize that the integrand is radially symmetric, so it is constant on spherical shells about the origin. We may then express the integral as an integral in one variable - $r$ the radius of the shells (similar to how in a second semester of calculus one might calculate the volume of a surface of revolution with cylindrical shells). Each shell contributes the value of the function on that shell times the surface area of the shell. Then we should get that our integral is equal to $$ \int_{0}^{1} r^{-m} S(r) \; dr, $$ where $S(r)$ is the surface area of the sphere of radius $r$.

I haven't checked this latter method, but I believe it agrees with the former one. Sorry if it doesn't work out.