What is the integral of $\int_0^{\sqrt y} e^{-x^2/2}.dx $?

Question

My teacher solved this by differential under integral sign method by a "formula" and brought the answer $e^{(\sqrt y)^2/2} \times 1/2\sqrt y$, given $y = x^2$ . I don't understand this. Please help.

Answer 1

We have $\int_{0}^{\sqrt{y}}e^{-\frac{x^{2}}{2}}dx=\sqrt{\frac{{\pi}}{2}}erf(\frac{\sqrt{y}}{\sqrt{2}})+C$ where $erf(x)$ is the error function.