[stargroup100]

One bit of insight as to how you can know these formulas can be derived from each other without even needing to do any work is knowing the basic relationships of functions and chain rule. For example, these two equations are related the same way you know that f(x) = x^2 is related to g(x) = 3(x-4)^2.

The transformations that turn the Gaussian Integral into the normal distribution are:
- Horizontal translation by a constant, μ (which obviously does not affect the integral given the limits follow the translation)
- Horizontal stretch by a constant factor, σ√2 (which changes the integral by that factor)
- Vertical stretch by a constant factor, 1/σ√(2π) (which changes the integral by that factor)

√π * σ√2 / σ√(2π) = 1