Difficult Gaussian Integral with Proof

The Gaussian integral is given by:

∫∞ -∞ e^(-x^2) dx = √π

Proof:

To prove this, we start by squaring the integral:

(∫∞ -∞ e^(-x^2) dx)^2 = ∫∞ -∞ ∫∞ -∞ e^(-(x^2 + y^2)) dx dy

Converting to polar coordinates, we have:

x = r cos(θ), y = r sin(θ), dx dy = r dr dθ

The integral becomes:

∫∫ e^(-(x^2 + y^2)) dx dy = ∫ from 0 to 2π ∫ from 0 to ∞ e^(-r^2) r dr dθ

Simplifying this, we get:

∫ from 0 to 2π ∫ from 0 to ∞ e^(-r^2) r dr dθ = 2π * (1/2) = π

Therefore, ∫∞ -∞ e^(-x^2) dx = √π