What is the integral of e^(2x^2)?

Aug 3, 2015

$\int {e}^{2 {x}^{2}} \mathrm{dx}$ cannot be expressed using elementary functions. You need the imaginary error function, erfi(x).

Explanation:

The imaginary error function is $\frac{2}{\sqrt{\pi}} \int {e}^{{x}^{2}} \mathrm{dx}$

$\int {e}^{2 {x}^{2}} \mathrm{dx}$ can be intergrated using substitution $u = \sqrt{2} x$ so $\mathrm{du} = \sqrt{2} \mathrm{dx}$ and we get:

$\int {e}^{2 {x}^{2}} \mathrm{dx} = \frac{1}{\sqrt{2}} \int {e}^{{u}^{2}} \mathrm{du}$

$= \frac{1}{\sqrt{2}} \frac{\sqrt{\pi}}{2} \text{erfi} u + C$

$= \frac{\sqrt{2 \pi}}{4} \text{erfi} \left(\sqrt{2} x\right) + C$