The following integral came up in another question:

$$ \int\limits_{0}^{x}\int\limits_{0}^{2x-x_1}\ldots\int\limits_{0}^{nx-x_1-x_2-\cdots-x_{n-1}}\mathrm dx_{n} \ldots \mathrm dx_1 $$

where $x>0$ is a real number and $n\in\mathbb{N}$. How should one go about this? Trying successive integration gets messy quickly.


This relates to something called the parking function. I would say the iterated integral is always equal to

$$ \int\limits_{0}^{x}\int\limits_{0}^{2x-x_1}\ldots\int\limits_{0}^{nx-x_1-x_2-\cdots-x_{n-1}}\mathrm dx_{n} \ldots \mathrm dx_1 = \frac{(n+1)^{n-1}}{n!}.x^n$$



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