Denote $x=(x_1,...,x_n)$. I want to calculate the following integral: $$\int_{S^{n-1}}x_1^2dS$$
Where $S^{n-1}$ is the $n$ dimensional sphere.
I was given a hint to use the fact that the volume of $S^{n-1}$ is $\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$, but I couldn't think of a way to use it.