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Integral proofs that $355/113>\pi$. (English) Zbl 1181.11077

Introduction: “One of the more beautiful results related to approximating $\pi$ is the integral

${\int }_{0}^{1}\frac{{x}^{4}{\left(1-x\right)}^{4}}{1+{x}^{2}}\phantom{\rule{0.166667em}{0ex}}dx=\frac{22}{7}-\pi ·\phantom{\rule{2.em}{0ex}}\left(1\right)$

Since the integrand is nonnegative on the interval $\left[0,1\right]$, this shows that $\pi$ is strictly less than $22/7$, the well known approximation to $\pi$. The first published statement of this result was in 1971 by D. P. Dalzell [Eureka 34, 10–13 (1971)], although anecdotal evidence [see J. M. Borwein, The life of Pi, history and computation, seminar presentation 2003, available from http://www.cecm.sfu.ca/~jborwein/pi-slides.pdf, March 2005] suggests it was known by Kurt Mahler in the mid-1960s. The result (1) is not hard to prove, if perhaps somewhat tedious. A partial fraction decomposition leads to a polynomial plus a term involving $1/\left(1+{x}^{2}\right)$, which integrates immediately to the required result. An alternative is to use the substitution $x=tan\theta$, leading to a polynomial in powers of $tan\theta$. We then apply the recurrence relation for taking the integrals of powers of $tan\theta$. Of course, the simplest approach today is to simply verify (1) using a symbolic manipulation package such as Maple or Mathematica.

An obvious question at this point might be whether similar elegant integral results can be found for other rational approximations for $\pi$. A particularly good approximation is 355/113, which is accurate to seven digits. Our aim here is to find a variety of such integral results.”

However, despite several variations of the style of integrand, no simple and elegant result was found.

The article is highly recommended as a basis of an undergraduate project.

##### MSC:
 11Y60 Evaluation of constants
##### Keywords:
approximation of pi
Maple