How to calculate the following integral: for positive constants $a_1, \cdots, a_{n+1}, $ and $i>0$ $ \int_{S^n\bigcap\{u_m\geq 0,\ m=1,\cdots, n+1\}}\left(\sum_{m=1}^{n+1} \frac{u_m}{a_m}\right)^{-i}du, $ where $u=(u_1, \cdots, u_{n+1})\in S^n,$ the unit sphere.
Thanks a lot!