Question

How do you find the integral of `e^(x^2)`?

Answer 1

Use some kind of approximation method. There is no nice, finitely expressible antiderivative.

(Other that to write: `int e^(x^2) dx`, of course.)

Answer 2

One symbolic way to do it is to use infinite series. Since `e^{x}=1+x+x^{2}/{2!}+x^{3}/{3!}+\cdots=1+x+x^{2}/2+x^{3}/6+\cdots` (for all `x`), it follows that `e^{x^{2}}=1+x^{2}+x^{4}/2+x^{6}/6+\cdots` (for all `x`).

It is valid in this example to now integrate term-by-term (the result is true for all `x`):

`\int e^{x^{2}} dx=\int (1+x^{2}+x^{4}/2+x^{6}/6+\cdots) dx`

`=C+x+x^{3}/3+x^{5}/10+x^{7}/42+\cdots`.

Alternatively, you can also give the antiderivative a name. Wolfram Alpha writes the antiderivative whose graph goes through the origin as `\frac{\sqrt{\pi}}{2}\mbox{erfi}(x)`, where `\mbox{erfi}(x)` is called the "imaginary error function".