Integral of $$$e^{i a x^{2}}$$$ with respect to $$$x$$$

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

Let $$$u=x \sqrt{i a}$$$.

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

So,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{\sqrt{i a}}$$$ and $$$f{\left(u \right)} = e^{u^{2}}$$$:

$${\color{red}{\int{\frac{e^{u^{2}}}{\sqrt{i a}} d u}}} = {\color{red}{\frac{\int{e^{u^{2}} d u}}{\sqrt{i a}}}}$$

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

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

Recall that $$$u=x \sqrt{i a}$$$:

$$\frac{\sqrt{\pi} \operatorname{erfi}{\left({\color{red}{u}} \right)}}{2 \sqrt{i a}} = \frac{\sqrt{\pi} \operatorname{erfi}{\left({\color{red}{x \sqrt{i a}}} \right)}}{2 \sqrt{i a}}$$

Therefore,

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

Add the constant of integration:

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

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