Suppose $S$ is the sphere of radius one. And suppose $f:S\to \mathbb{R}$ is defined as follows: $$f(x_1,...,x_n)=\frac{1}{x_1^2}+\frac{1}{x_2^2}+\cdots+\frac{1}{x_n^2}$$
I am trying to calculate either of these: $$\int f(\gamma)d\gamma$$ where integrating is with respect to Haar measure over the sphere. Or maybe finding the median of the induced random variable on $\mathbb{R}$ by $f$. I appreciate any help in this direction.