Integral of $$$e^{- x^{2}}$$$

Find $$$\int e^{- x^{2}}\, dx$$$.

This integral (Error Function) does not have a closed form:

$${\color{red}{\int{e^{- x^{2}} d x}}} = {\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}\right)}}$$

Therefore,

$$\int{e^{- x^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}$$

Add the constant of integration:

$$\int{e^{- x^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2}+C$$

$$$\int e^{- x^{2}}\, dx = \frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{2} + C$$$A