How to calculate an integral with an unknown number of integration variables?

Question

How to calculate the following integral, which has an unknown number of integration variables? $$ \int\limits_{-\infty}^{+\infty}\cdots\int\limits_{-\infty}^{+\infty}\exp\left[-\dfrac{1}{2\theta}\sum_{i=1}^n\big(b_i-\sum_{j=1}^ma_{ij}x_j\big)^2-\dfrac{1}{2(1-\theta)}\sum_{j=1}^mx_j^2\right]dx_1,\dots,dx_m $$ where $\theta\in(0,1)\ \text{and}\ a_{ij},b_i,x_j\in(-\infty,+\infty)$.

Answer 1

$$\int_{R^m}e^{-\frac{1}{2}x^T\Sigma^{-1}x+\langle s, x \rangle }dx=(\sqrt{2\pi})^m e^{\frac{1}{2}s^T\Sigma s}.$$ Apply this to

$$\Sigma^{-1}=\frac{1}{1-\theta}I_m+\frac{1}{\theta}A^TA, \ \ \ s=\frac{1}{\theta}A^Tb.$$