I need help to proof the following: Consider the multiple integrals below:
$$ \int_{a}^{b} dx_{1} \int_{a}^{b} dx_{2} \cdots \int_{a}^{b} dx_{n} \ f(x_{1}, x_{2}, ..., x_{n}).$$
Here, $f$ is a function that is unchanged if we exchange two of your variables, that is
$$f(..., x_{i}, ..., x_{j}, ...) = f(..., x_{j}, ..., x_{i}, ...).$$
I have to prove that this integral is equal to
$$n! \int_{a}^{b} dx_{1} \int_{a}^{x_{1}} dx_{2} \cdots \int_{a}^{x_{n-1}} dx_{n} \ f(x_{1}, x_{2}, ..., x_{n}).$$
It may look likes weird, but I am studying many-body perturbation theory using Feynman diagrams and this relation is very useful. I know that this is true, but I can't find anywhere a good proof or justification for it.
You are integrating over the region where $ b \geq x_1 \geq x_2 \geq x_3 \geq \ldots \geq x_n \geq a $.
By the condition that the function is symmetric in the variables, it follows that this is $\frac{1}{n!}$ of the original integral. Hence the equality holds.