Let $M = \{ x \in \mathbb{R}^3| |x|^2 = 2<x,a> -\frac{3}{4}|a|^2 \}$ where $a \in \mathbb{R}^3$ is some constant vector such that $|a| \neq 0$.
I want to calculate the integral $\int_M \frac{1}{|x|} ds$.
First, $M$ is a smooth manifold, and is actually the sphere around the point $(a_1, a_2, a_3)$ with the radius $\frac{1}{2}|a|$. (Here I denote $a = (a_1, a_2, a_3)$)
I tried using the integral straight forward using a parametrization, but the expression in the integral becomes not very nice.
I also thought of somehow using the Divergence theorem, since the function $f(x) = \frac{1}{|x|}$ is harmonic. So if I could use the field $\nabla f$ it would be over. But the outer normal vector here is $2\frac{x-a}{|a|}$, and I don't really know how work with it.
Help would be appreciated.