Question

∫e^x^2 dx=?

Answer 1

`\int e^(x^2)dx = sqrt(pi)/2 "erfi"(z)`

Explanation:

If you were to modify this integral slightly:

`\int e^(x^2)dx -> color(blue)((sqrt(pi))/(2))*(2)/(sqrt(pi)) \int e^(x^2)dx`

This now resembles the imaginary error function `"erfi"(z)`, where `z` is an uniformly distributed random variable.

`=> (2)/(sqrt(pi)) \int_0^z e^(x^2)dx = "erfi"(z)`

Hence:
`\int e^(x^2)dx = color(blue){sqrt(pi)/2}"erfi"(z)`

This integral comes up in quantum mechanics due to its statistical nature. If you want more information on the error function, see this .