In this post I want to show this nice integral I saw once during my graduate studies.
The cool thing about it is the integral’s value is independent of . The shape of the integrand changes as is varied but the area under the curve remains the same.
To prove this, we denote the left side as (a continuous function of ) and differentiate under the integral sign.
The interchange of integral and derivative is valid here as the integrand and its derivative are continuous in the open interval of interest, plus the integral exists. Next, note that if we make the substitution then and
Note that this is not valid when . Combining (1) and (2),
Hence the integral is independent of in the region where it is differentiable as a function of . Hence it is equal to its constant value in the limit , which is the Gaussian integral
as we wished to show. Note that when we can replace with in the above and find that for
which though no longer constant has the same limit as .