I am interested in the asymptotics of the integral
$$I(a):=\int\_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$
for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\lesssim e^{-a^2}$ for large $a$. Numeric integration indicates that
$$ \lim\_{a\to\infty} I(a) e^{a^2} a^{5/2} =C \in(0,1)$$
for some constant $C$.


Are there integral tables where I could obtain explicit expressions for $I(a)$ for finite $a$? If not, are the asymptotics above correct, and what is $C$?


So far I could only find results when the error function is integrated against integer powers of $x$ e.g. in Section 2.14 of [https://intra.ece.ucr.edu/~korotkov/papers/Korotkov-book-integrals.pdf](https://intra.ece.ucr.edu/%7Ekorotkov/papers/Korotkov-book-integrals.pdf)



Asymptotics of error function integral with square root
$$\int\_0^\infty \sqrt{x}\,\mathrm{Erfc}(x+a)\,\mathrm{d}x\rightarrow \frac{e^{-a^2}}{(2a)^{5/2}},\;\;\text{for}\;\;a\rightarrow \infty,$$
so $C=2^{-5/2}.$ Corrections are smaller by a factor $1/a$.  

I obtained this asymptotics by integrating the large-$a$ expansion of the error-function,
$$\text{Erfc}\,(x+a)\rightarrow \frac{e^{-(a+x)^2}}{\sqrt{\pi } a},\;\;\text{for}\;\;a\rightarrow \infty.$$


