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LSpice
LSpice
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Closed form for $ \int_{0}^{1} \cdots\dotsi \int_{0}^{1} \frac{x_1^q + \cdots\dotsb + x_n^q}{x_1^p + \cdots\dotsb + x_n^p} \, \mathrm{d}x_1 \cdots\dotsm \mathrm{d}x_n $

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

Recently, reading this problemthis problem, I found out that

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

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

Closed form for $ \int_{0}^{1} \cdots \int_{0}^{1} \frac{x_1^q + \cdots + x_n^q}{x_1^p + \cdots + x_n^p} \, \mathrm{d}x_1 \cdots \mathrm{d}x_n $

I asked this question here, but received no answer.

Recently, reading this problem, I found out that

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

Is it known a general formula 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?

Closed form for $ \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 $

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?

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user967210
user967210
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Closed form for $ \int_{0}^{1} \cdots \int_{0}^{1} \frac{x_1^q + \cdots + x_n^q}{x_1^p + \cdots + x_n^p} \, \mathrm{d}x_1 \cdots \mathrm{d}x_n $

I asked this question here, but received no answer.

Recently, reading this problem, I found out that

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

Is it known a general formula 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?