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

2 Answers
Apr 27, 2015

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

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

Apr 29, 2015

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".