(FB/Math/BM – Question by Pål Tingsbø)
Is it possible to calculate e from an integral?
I suppose you mean an integral where e is not to start with in the integrand or the integration limits. But while π occurs in the values of countless integrals it seems hard to find e. The best I have found at present is
∫ [-∞,∞] (cos x)/(1 + x²) dx = π/e.
The integrand is not quite ”e-free” since cos is closely related to the exponential function, at least in the clomplex domain. Also to be useful to calculate digits in e the integral must be numerically calculated.
Why is it that π is so common but e so uncommon as values of integrals unless it is there to begin with. That π is common can be traced to the fact that the derivative of arctan x is a rational function and has value π/4 for a ”simple” x, x = 1. But Dex = ex so nothing similar happens for e. That’s one start for an explanation maybe.