Consider the following real integral $$ \int_{a}^b dx~ w(x)\exp[-N f(x)] $$ and assume that $f(x)$ is twice-differentiable in $(a,b)$, and there exists a unique minimum point $x^\star\in (a,b)$ for $f$ - i.e. $f'(x^\star)=0$ and $f''(x^\star)>0$.
The standard Laplace approximation states that the integral can be approximated as $$ \approx w(x^\star)\exp[-N f(x^\star)]\sqrt{\frac{2\pi}{Nf''(x^\star)}} . $$
But what if $w(x^\star)=0$? How should the "classical" formula above be modified?
For example, we know by exact evaluation that $$ \int_{-\infty}^3 dx~ x \exp[-N x^2]=-\frac{e^{-9N}}{2N} . $$ But how can I retrieve the expression on the r.h.s. (which is already the correct asymptotic expansion) using the Laplace approximation? The argmin of the function $x^2$ in the exponent is at $x=0$, but this value will kill the function $w(x)=x$....