Integral of $$$の e^{- x^{2}}$$$ with respect to $$$x$$$

Find $$$\int の e^{- x^{2}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=の$$$ and $$$f{\left(x \right)} = e^{- x^{2}}$$$:

$${\color{red}{\int{の e^{- x^{2}} d x}}} = {\color{red}{の \int{e^{- x^{2}} d x}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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