A note on a certain non-Gaussian integral

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In this paper we present a general formula for the inhomogeneous non-Gaussian integral Id(S₁,S₂)=∫ dx₁... dxd e-1/2S₁²-S₂, where S₁ and S₂ are symmetric quadratic forms. The solution depends on the eigenvalues of the matrix A=-iM₂M₁⁻¹, where M₁ and M₂ are the matrix representations of S₁ and S₂ 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(S₁,S₂) in the infinite-dimensional case is calculated and the solution depends only on the determinants of M₁ and M₂. The infinite-dimensional case may be of use in QFT.