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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!

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Have you tried evaluating it for some small $n$? Do you think the method can be generalized to any $n$? – Pedro Tamaroff Jun 15 '12 at 2:30
Try it out of some values, may is not enough to be sure. – Pedro Tamaroff Jun 15 '12 at 2:56
The evaluating process may be the same for any $n.$ In fact, I need it may less than $$C(\frac{a_1\cdots a_{n+1}}{a_2\cdots a_{n+1}})^i$$ if $a_1\leq\cdots\leq a_{n+1}.$ – mathpdE Jun 15 '12 at 2:58

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