Gaussian-like integral on spherical hypersurface

Question

I am interested in the calculation of the following integral in $\mathbb R^N-1$: \begin{equation} I_A=\int_{||x||^2=R^2} \mathrm d \sigma \; e^{-\frac 12 x^T A x} \end{equation} where $A$ is postive definite. I tried to rewrite the surface of integration as a delta function \begin{equation} \delta(||x||^2-R^2) = \frac 1 {2\pi}\int_{-\infty}^{+\infty} \mathrm d k\; e^{ik(||x||^2-R^2)} \end{equation} and to incorporate the above constraint in the precision matrix, by rescaling \begin{equation} A \rightarrow B(k)= A-2ik1_N \end{equation} where $1_N$ denotes the $N\times N$ identity matrix. I cannot however progress further with standard results on Gaussian integrals, given that $\Re(\det B(k))<0$ for large values of $k$. Is there another approach to solve $I_A$?