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

Recently, reading this problem, 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?