Asymptotic growth of $\int_{-2c}^0\frac{\exp(-x^2)}{x+c}dx$

Question

I need to estimate the growth of $$ \int_{-2c}^0\frac{\exp(-x^2)}{x+c}dx $$ as $c\to\infty$. In particular I want to show that $$ \int_{-2c}^0\frac{\exp(-x^2)}{x+c}dx=\mathcal{O}\left(\frac{1}{c}\right) $$ How should I proceed? Do you have any hint?

Answer 1

Due to singularity at $x=-c$, this must be considered in the sense of principal value. For the estimation, just write $\int_{-2c}^{0}=\int_{-2c}^{-c/2}+\int_{-c/2}^{0}$. The first is exponentially decaying with $c$ (not hard to check), the second is trivially bounded from above (with $\frac{2}{c}\int_{-\infty}^{0}\exp(-x^2)\,dx$) and, if desired, can be similarly bounded from below (with $\frac{1}{c}\int_{-c/2}^{0}\exp(-x^2)\,dx$).

To go more precise, this integral is $\displaystyle\frac{1}{c}\int_{0}^{c}\frac{e^{-t^2}-e^{-(2c-t)^2}}{1-t/c}\,dt\thicksim\frac{\sqrt\pi}{2c}$.