Consider the Hamiltonian of Ising model
$$H= - \sum_{x,y} J_{x,y}\sigma_x \sigma_y - \sum_x h\sigma_x$$
where $\sigma_x$ can be $1$ or $-1$ and the number of sites is finite ( i.e. there are a finite number of $x,y$ ) and where $x,y$ are points of the a $d$-lattice with side $L$ (i.e. $x,y \in \mathbb{Z} \cap[-L/2,L/2].$ So the partiction function is $$Z=\sum_{\sigma}e^{-\beta H}=\sum_{\sigma}e^{\beta\sum_{x,y} J_{x,y}\sigma_x \sigma_y} e^{\sum_x h\sigma_x}$$ where the sum is over all the possible choice of $\sigma=\pm 1$.
The goal is to write the partiction function as a Gaussian integral using the following formula$$\int e^{-\frac{1}{2}\sum\limits_{i,j=1}^{n}A_{ij} x_i x_j+\sum\limits_{i=1}^{n}B_i x_i} d^nx=\int e^{-\frac{1}{2}\vec{x}^T \mathbf{A} \vec{x}+\vec{B}^T \vec{x}} d^nx= \sqrt{ \frac{(2\pi)^n}{\det{A}} }e^{\frac{1}{2}\vec{B}^{T}\mathbf{A}^{-1}\vec{B}}$$ in view of the fact that $e^{\beta\sum_{x,y} J_{x,y}\sigma_x \sigma_y}$ has the form $e^{\frac{1}{2}\vec{B}^{T}\mathbf{A}^{-1}\vec{B}}$ ( thinking $\sigma=(\sigma_{x_1},...)$ as a vector with $L^d$ components and $J=(J_{xy})$ as a matrix $L^d\times L^d$) .
Now, what i read on the notes is that
$$e^{\beta\sum_{x,y} J_{x,y}\sigma_x \sigma_y}= (2\pi)^ {L^d}(detJ)^{-1/2}\int_{-\infty}^{+\infty}\prod_{x} d \varphi_x e^{-\frac{1}{2\beta}\sum_{x,y}\varphi_x J^{-1}_{xy}\varphi_y + \sum_{x}\varphi_x \sigma_x}.$$
To verify the last formula I tried to calculate the integral in the right head side using the classical techniques for the Gaussian integral ( completion of the square), but I didn't found the last formula. So my question is: is there something wrong with the last expression for $e^{\beta\sum_{x,y} J_{x,y}\sigma_x \sigma_y}$ as Gaussian integral?