I'm trying to find a closed form expression for the following integral:
$$ \frac{1}{\sqrt{2\pi\sigma^2}}\int_0^{\infty} \sqrt{x} \exp \bigg(\frac{-(x-\mu)^2}{2\sigma^2} \bigg)dx$$
This integral is essentially $E[\sqrt{x} \mathbf{1}_{x>0}]$, where $X\sim\mathcal{N}(\mu,\sigma^2)$