Looking at an expression for $\pi$

$$\pi = \frac{22}7-\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx$$

it seems to me that the integral expression is the error between the approximation $\frac{22}7$ and $\pi$. Is there any history behind the discovery of this integral? Was it tied to the error between the two numbers or was it discovered separately and only later applied?

  • $\begingroup$ Related (but not the answer to this exact question): math.stackexchange.com/questions/1079024/… I guess someone playing around with integrals of this form stumbled on this relationshihp. $\endgroup$
    – Simon S
    Commented Mar 11, 2015 at 18:25
  • 1
    $\begingroup$ I think it was on the 1968 Putnam exam, so it's older than that. Here is a Wikipedia article about this equality. ${}\qquad{}$ $\endgroup$ Commented Mar 11, 2015 at 18:25
  • $\begingroup$ @MichaelHardy Thanks for the Wikipedia article but it doesn't really explain where the integral as the exact error came from. $\endgroup$
    – Gabriel
    Commented Mar 11, 2015 at 18:29

1 Answer 1


This is the earliest instance of the formula that I could find.


J. London Math. Soc. (1944) 19 (75 Part 3): 133-134. doi: 10.1112/jlms/19.75_Part_3.133

Perhaps this is where it was first discovered.

  • $\begingroup$ I can't access it; could you upload a picture or whatever is in the document please? $\endgroup$
    – Gabriel
    Commented Mar 11, 2015 at 21:19

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