A note on a certain non-Gaussian integral

In this paper we present a general formula for the inhomogeneous non-Gaussian integral $I_d(S_1,S_2)=\int dx_1... dx_d e^{-{1/2}S_1^2-S_2}$, where $S_1$ and $S_2$ are symmetric quadratic forms. The solution depends on the eigenvalues of the matrix $A=-iM_2M_1^{-1}$, where $M_1$ and $M_2$ are the matrix representations of $S_1$ and $S_2$ respectively. In the 2-dimensional case we also give a manifestly SO(2)-invariant formulation in terms of invariants of the matrix $A$. An expression for $I(S_1,S_2)$ in the infinite-dimensional case is calculated and the solution depends only on the determinants of $M_1$ and $M_2$. The infinite-dimensional case may be of use in QFT.