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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.)

Inquitor
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  • How about $\int_0^e 1dx$? – vadim123 Mar 08 '15 at 19:46
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    "Obvious" is vague; every connection is obvious after it's made and explained. I suggest to make the question more precise: the integrand must be an algebraic function with rational coefficients, and the limits of integration must be rational. –  Mar 08 '15 at 20:49

3 Answers3

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$e$ 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

marwalix
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$\int_{-\infty}^1 \sum_{k=0}^\infty {1 \over k!}t^k dt$

copper.hat
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  • It seems obviously connected to $e$. – Emilio Novati Mar 08 '15 at 19:20
  • Picky. ${}{}{}{}$ – copper.hat Mar 08 '15 at 19:22
  • What I am looking for, ideally, is something not to involve infinite series, of even infinite as the integration limit. An integrand that will represent an area easy to visualize, and preferably not going to infinite. My apologies, for not specifying this above. – Inquitor Mar 08 '15 at 19:22
  • Interesting question! The log, in the sense you describe, is "natural," and it is therefore not surprising that it was discovered first. – André Nicolas Mar 08 '15 at 19:27
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Not visualizable as an area, but $$ 2^{\int_{0}^{1}2^x dx} $$ is nice.

mjqxxxx
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