I want a numerical method to evaluate:
$$\log \int_a ^b f(x) \mathrm{d}x$$
when what I have is a numerical routine to evaluate $\log f(x)$. The problem is that if $f(x)$ takes very large or very small values, direct evaluation of $f(x)$ will lead to overflows/underflows. I am wondering if anyone has approached this numerical integration problem in general?
Note that even though $f(x)$ can be very small in some regions and very large in other regions, you can't just approximate the small values to zero, because if the region where $f(x)$ is very small is large enough, it may contribute significantly to the integral.