Infinite Gaussians

The Bonn Integration Bee is a yearly competition where people try to manually solve integrals. For the first quarterfinal of the 2025 edition, contestants had to solve the indefinite integral

whose solution is

Upon plotting this equation, I noticed the function looks very similar to a series of Gaussian peaks, periodic in π:

Indeed, the two functions are equal to within <0.1%, with A = 0.595860 and σ = 0.403627, found numerically. The main discrepency between the two functions is at integer multiples of π, where the trigonometric equation evaluates to zero, whereas the sum of Gaussians will always have some non-zero value at these points.

This function can be regarded as a Gaussian-broadened Dirac comb, with periodicity π. Now, given that this sum of Gaussians is approximately equal to the indefinite integral of the original trigonometric function, this implies that there is also a function whose indefinite integral is a sum of Gaussian, and which should also closely match the original trig function. We find this by differentiating the sum of Gaussians:

The correspondence can be easily seen visually.

Note that the periodicity can be changed by replacing the .pi. in the sum of Gaussian by T, and by dividing the arguments of the sine and cosine functions by the same.