I am considering calculating the following definite integral. The limits of integration are positive and $c > 0$. One can see that the integrand is bounded above by the Gaussian in the numerator and so the integral should converge
$$ \int_{a}^b \frac{e^{-cx^2}}{x+1}dx $$
Suppose the limits of integrations are several standard deviations (of $e^{-cx^2}$) away. Then it makes sense to approximate $e^{-cx^2}$ with some function that models the Gaussian's end behaviors quite well. The integral then seems tractable to me. For instance, if the numerator was approximately $1/cx$, then I could use partial fraction decomposition and natural logarithms.
What is the functional form of the Gaussian $e^{-cx^2}$ at its tails?