The function e^(−x²) is the bell curve: it peaks at x=0, falls to zero in both directions, and underlies the normal distribution in statistics. Its integral over all of ℝ is exactly √π ≈ 1.7724. This is remarkable: e and π, usually encountered separately, are linked in the most fundamental integral of statistics.
The standard proof is beautiful: let I = ∫e^(−x²)dx. Compute I² = (∫e^(−x²)dx)(∫e^(−y²)dy) = ∫∫e^(−(x²+y²))dxdy, then switch to polar coordinates. The double integral becomes ∫₀^∞ e^(−r²) · 2πr · dr = π. So I² = π and I = √π.
The normal distribution N(0,1) has density (1/√(2π))e^(−x²/2). The factor 1/√(2π) is exactly what's needed for the total probability to equal 1, and it comes from the Gaussian integral. Statistics, quantum mechanics, and information theory all rest on this single result.