Derive generalized Gaussian Integral from the Special Case of Gaussian Integral for Mu = 0 and Sigma = 1-by-sqrt(2)

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What is the integral 'I' with respect to Gaussian distribution?

The integral with respect to a Gaussian distribution, often referred to as the Gaussian integral, is a classical mathematical result. The Gaussian integral is defined as follows:

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

This integral is frequently encountered in probability theory and statistics, especially when dealing with normal (Gaussian) distributions. The integral represents the area under the bell-shaped curve of the standard normal distribution.

It's important to note that the Gaussian integral has a well-known result, but proving it involves more advanced mathematical techniques such as completing the square and complex analysis. The result is widely used in various branches of mathematics, physics, and engineering. If you encounter a Gaussian distribution with mean $\mu$ and variance $\sigma^2$, the integral can be generalized as:

$\int_{-\infty}^{\infty} e^{-(x-\mu)^2/(2\sigma^2)} \,dx = \sqrt{2\pi\sigma^2}$

This generalization involves a change of variable and scaling to account for the mean ($\mu$) and variance ($\sigma^2$).

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Fig 2:



Showing how Generalized Gaussian Integral and Special Case of Gaussian Integral are related:
Fig 3: