multivariate gaussian integral with linear constraint

Question

I am interested in the calculation of the multivariate gaussian integral \begin{equation*} \int_{\mathbb R^N} \mathrm d x_1\cdots \mathrm d x_N \; \prod_{j=1}^Ne^{-\frac{x_j^2}{2\sigma_j^2}}\delta\left(x-\sum_{l=1}^N c_l x_l\right) \end{equation*} for a set of $c_l\in\mathbb R$. Is there a way to solve the integral with this linear constraint? If so, how? As a simple separation of variables does not apply anymore, I would guess that a smart change of variables could be a way to tackle the calculation.

Answer 1

Typically one uses the representation $\delta(f(x))={1\over 2\pi}\int dk e^{i k f(x)}$