TITLE: Is there a definite integral that evaluates to the constant $e$?
QUESTION [6 upvotes]: The integrand should not involve the constant $e$ itself nor, preferably, $\cosh$, $\sinh$, etc. $\pi$ arises in definite integrals such as
$$\int_0^a \frac{dx}{\sqrt{a^2-x^2}} = \frac{\pi}{2}$$
The integrand must be an algebraic function with rational coefficients, and the limits of integration must be rational. (Thank you Woodface for the suggestion.)

REPLY [8 votes]: $e$ is not a period i.e. not a number that can be represented by the integral of a rational or irrational function over a domain defined by rational functions. The periods form a subring of $\mathbb{C}$. There is a very good article by D. Zagier and M.Kontsevitch on periods. Don't have the reference but Google should help