Integral of $$$e^{3 x^{2}}$$$

Find $$$\int e^{3 x^{2}}\, dx$$$.

Let $$$u=\sqrt{3} x$$$.

Then $$$du=\left(\sqrt{3} x\right)^{\prime }dx = \sqrt{3} dx$$$ (steps can be seen »), and we have that $$$dx = \frac{\sqrt{3} du}{3}$$$.

The integral becomes

$${\color{red}{\int{e^{3 x^{2}} d x}}} = {\color{red}{\int{\frac{\sqrt{3} e^{u^{2}}}{3} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{\sqrt{3}}{3}$$$ and $$$f{\left(u \right)} = e^{u^{2}}$$$:

$${\color{red}{\int{\frac{\sqrt{3} e^{u^{2}}}{3} d u}}} = {\color{red}{\left(\frac{\sqrt{3} \int{e^{u^{2}} d u}}{3}\right)}}$$

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

$$\frac{\sqrt{3} {\color{red}{\int{e^{u^{2}} d u}}}}{3} = \frac{\sqrt{3} {\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erfi}{\left(u \right)}}{2}\right)}}}{3}$$

Recall that $$$u=\sqrt{3} x$$$:

$$\frac{\sqrt{3} \sqrt{\pi} \operatorname{erfi}{\left({\color{red}{u}} \right)}}{6} = \frac{\sqrt{3} \sqrt{\pi} \operatorname{erfi}{\left({\color{red}{\sqrt{3} x}} \right)}}{6}$$

Therefore,

$$\int{e^{3 x^{2}} d x} = \frac{\sqrt{3} \sqrt{\pi} \operatorname{erfi}{\left(\sqrt{3} x \right)}}{6}$$

Add the constant of integration:

$$\int{e^{3 x^{2}} d x} = \frac{\sqrt{3} \sqrt{\pi} \operatorname{erfi}{\left(\sqrt{3} x \right)}}{6}+C$$

$$$\int e^{3 x^{2}}\, dx = \frac{\sqrt{3} \sqrt{\pi} \operatorname{erfi}{\left(\sqrt{3} x \right)}}{6} + C$$$A