Gaussian Integral: Integrate exp(-x^2) from -infinite to +infinite

Integrate exp(-x^2) from -infinite to +infinite

Note: this integral is also known as Gaussian integral.

The integral of $e^{-x^2}$ from $-\infty$ to $+\infty$ is a well-known mathematical constant and is denoted by $\sqrt{\pi}$. The integral is often expressed as:

$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$

This result is commonly encountered in probability theory and statistics, especially in the context of the normal distribution. The integral doesn't have a simple elementary function representation and is typically solved using advanced mathematical techniques. The proof involves the use of polar coordinates and complex analysis.

Some of the mathematical results and properties we will use today:

Note (1):

I = Integral of f(x) w.r.t. dx
Then I^2 = Integral of f(x).f(y) w.r.t. dxdy

Note (2): Well known step:

When changing to polar coordinates from (x, y) to (r, theta):

# (dx dy) changes to (r dr dtheta) 

Note (3): Well known trigonometric property:

sq(sin(theta)) + sq(cos(theta)) = 1

Note (4):

(integral of e^(x)) = e^(x)

The result that folows from this is: (integral of e^(-x)) = -e^(-x)

Fig 1:


Fig 2: