Integral 0 to infinity e^(-x2)dx ? Here's a nice trick..

Vishal U. Karve

Why?

Well, there isn't a closed-form expression for the antiderivative of the integrand, so the Fundamental Theorem of Calculus won't help.

But the expression is meaningful, since the it represents the area under the curve from 0 to infinity.

Furthermore, there is a nice trick to find the answer!

Call the integral I. Multiply the integral by itself: this gives

I² =

then view as an integral over the first quadrant in the plane:
I²

then change to polar coordinates (!):

Now this is quite easy to evaluate: you find that

.

This means that I, the original value of the integral you were looking for, is . Wow!

Source:

http://www.math.hmc.edu/funfacts/ffiles/20008.3.shtml