Suppose $f(x)$ and $g(x)$ are a real-valued rapidly decaying functions that smoothly approach $1$ as $x\rightarrow 1$. For example,
$$f(x)=e^{-x}, \,g(x)=\frac{1}{1+x^2}$$
Now consider the following function defined through an integral:
$$F(\alpha)=\int_0^{\infty}\frac{f(x)}{x+\alpha g(x)}dx$$
Suppose you would like to study the small-$\alpha$ behavior of $F(\alpha)$. You can clearly see that if $\alpha=0$, the integrand is logarithmically singular near $x=0$. Also, if you analytically continue $\alpha$, it is clear that $F(\alpha)$ is singular along the entire negative-real axis.
Therefore, I expect the analytic small-$\alpha$ behavior to look like a logarithm
$$F(\alpha)\approx F_0\textrm{Log}(\alpha)+\mathcal{O}(\alpha\,\textrm{Log}\alpha)$$
Is there a name for integrals defined like $F(\alpha)$? Or maybe something similar?