Let $\vec{x} = (x_1, x_2, \dots, x_n) \in [0,1]^n$ and $f(\vec{x}): [0,1]^n \to \mathbb{C}$ be a function in these variables.
Is there a recursive scheme for this iterated integral?
$$\int_{[0,1]^n} \prod dx_i \;f(\vec{x}) $$
If $n = 10$ and I break $[0,1]$ into 100 segments, we have $10^{20}$ points to add up. There must be a smarter way.
In fact, the function I wish to integrate is the Haar measure of the Unitary group.
$$\int_{U(n)} f(A) \ dA = \frac{1}{n!} \int_{[0,2\pi]^n} \prod_{j<k} \big|e^{i \theta_j} - e^{i \theta_k}\big|^2 \cdot f(\theta_1, \ldots, \theta_n) \ \frac{d \theta_1}{2\pi} \ \cdots \ \frac{d \theta_n}{2\pi}$$