Integrate e^-x^2

e^-x^2 is not integrable directly, so a transformation has to first be made. Assume J = int(e^-x^2 dx), and x=y. Therefore:

J^2 = int(e^-x^2 dx * e^-y^2 dy)
J^2 = int(e^-(x^2+y^2) dx dy)

Transform:

J^2 = int(e^-r^2 rdr) * int(d(theta))

Usually, at this point, various sources integrate it from -inf to inf, and obtain sqrt(pi) as the solution of int(e^-x^2 dx) [-inf,inf].

My question is, why should this only work unless the function is integrated from -inf to inf? Why can't it work for any value of x or r?

Because you still have to evaluate this function (Note that you still have the in there, which you in the beginning wanted to transform to some other expression):


As r goes toward infinity and negative infinity, goes toward zero. This isn't true for real numbers, and thus we have an issue.

What if I wanted to evaluate the integral from 0 to 1?

Use the error function, or extrapolate it yourself. I recommend the error function.

Noah