# Strange $n$-dimensional volume integral

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.

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