How do you find the integral of ex2?

2 Answers
Apr 27, 2015

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

(Other that to write: ex2dx, of course.)

Apr 29, 2015

One symbolic way to do it is to use infinite series. Since ex=1+x+x22!+x33!+=1+x+x22+x36+ (for all x), it follows that ex2=1+x2+x42+x66+ (for all x).

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

ex2dx=(1+x2+x42+x66+)dx

=C+x+x33+x510+x742+.

Alternatively, you can also give the antiderivative a name. Wolfram Alpha writes the antiderivative whose graph goes through the origin as π2erfi(x), where erfi(x) is called the "imaginary error function".