[math] \newcommand{\mathds}{\mathbb}[/math]

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Establish the following integration formula over [math]S^{N-1}_\mathbb R\subset\mathbb R^N[/math], with respect to the normalized measure, valid for any exponents [math]p_i\in\mathbb N[/math],

[[math]] \int_{S^{N-1}_\mathbb R}|x_1^{p_1}\ldots x_N^{p_N}|\,dx=\left(\frac{2}{\pi}\right)^{\Sigma(p_1,\ldots,p_N)}\frac{(N-1)!!p_1!!\ldots p_N!!}{(N+\Sigma p_i-1)!!} [[/math]]

where [math]\Sigma=[odds/2][/math] if [math]N[/math] is odd and [math]\Sigma=[(odds+1)/2][/math] if [math]N[/math] is even, where “odds” denotes the number of odd numbers in the sequence [math]p_1,\ldots,p_N[/math].