Question

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

Answer 1

`int e^(2x^2) dx` cannot be expressed using elementary functions. You need the imaginary error function, erfi(x).

Explanation:

The imaginary error function is `2/sqrtpi int e^(x^2) dx`

`int e^(2x^2) dx` can be intergrated using substitution `u = sqrt2 x` so `du = sqrt2 dx` and we get:

`int e^(2x^2) dx = 1/sqrt2 int e^(u^2) du`

`= 1/sqrt2 sqrtpi/2 "erfi" u +C`

`= sqrt(2pi)/4 "erfi"(sqrt2x) +C`