I asked this question [on MSE][1], but received no answer.

Recently, reading [this problem][2], I found out that

$$ \lim_{n\to \infty} \int_{0}^{1} \dotsi \int_{0}^{1} \frac{x_1^q + \dotsb + x_n^q}{x_1^p + \dotsb + x_n^p} \, \mathrm{d}x_1 \dotsm \mathrm{d}x_n
=\frac{p+1}{q+1}. $$

Is a general formula known for that multiple integral when we fix $ n$, $p $ and $q$ as positive integers? Otherwise is it possible to have a general formula if $q=2$, $p=1$ and $n$ is a fixed positive integer?


  [2]: https://math.stackexchange.com/questions/4801426/lim-n-to-infty-int-01-cdots-int-01-fracx-1qx-2q-cdotsx-nqx-1px "\$\lim_{n\to+\infty}\int_0^1\cdots\int_0^1\frac{x_1^q+x_2^q+\cdots+x_n^q}{x_1^p+x_2^p+\cdots+x_n^p}dx_1dx_2\cdots dx_n=\frac{p+1}{q+1}\$"


  [1]: https://math.stackexchange.com/questions/4906414/general-formula-for-int-01-cdots-int-01-fracx-1q-cdots-x