The constant e and integrals

(FB/Math/BM – Question by Pål Tingsbø)

Is it possible to calculate e from an integral?

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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.

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