I need help calculating the following integral on a sphere:
$$\int_{S^{n-1}}\ x_1 x_2 dS\left(x\right)$$
I tried using my assumption (not totally sure it's correct) that $\int_{S^{n-1}} \ {x_1}^2 = \frac{2\pi^\frac{n}{2}}{n\Gamma\left(\frac{n}{2}\right)}$ to say that $$ 2\int_{S^{n-1}}\ x_1 x_2 dS\left(x\right) + \int_{S^{n-1}} \ {x_1}^2 + \int_{S^{n-1}} \ {x_2}^2 = \int_{S^{n-1}} \ \left({x_1}+{x_2}\right)^2$$
Is this the way to go? If it is, how to proceed from now on? if not - what's wrong? (better yet - what's right? )