= Solution
Because $A$ is a real symmetric <positive-definite matrix>, its inverse exists and all its <eigenvalues> are positive. <Completing the square> gives
$$
-\frac12u^TAu+b^Tu
=-\frac12(u-A^{-1}b)^TA(u-A^{-1}b)+\frac12b^TA^{-1}b.
$$
The translation $v=u-A^{-1}b$ has unit <Jacobian determinant>. Choose an <orthogonal matrix> $O$ with $O^TAO=\operatorname{diag}(a_1,\ldots,a_n)$, where $a_i>0$. The change $v=Oz$ also has absolute Jacobian one. The remaining integral is the product of $n$ one-dimensional <Gaussian integrals>, each equal to $\sqrt{2\pi/a_i}$. Consequently the <linear-source multivariate Gaussian integral> is
$$
\boxed{\int_{\mathbb R^n}d^nu\,e^{-u^TAu/2+b^Tu}
=\frac{(2\pi)^{n/2}}{\sqrt{\det A}}
\exp\left(\frac12b^TA^{-1}b\right).}
$$
The determinant square root is the positive one, as required by convergence of the original real integral.
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