The following integral,

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

is clearly positive, which proves that $\pi < 22/7$.

Is there a similar integral which proves $\pi > 333/106$?


This integral would do the job:

$$\int_0^1 \frac{x^5(1-x)^6(197+462x^2)}{530(1+x^2)}\:dx= \pi -\frac{333}{106}$$


Also you can refer to S.K. Lucas Integral proofs that $355/113 > \pi$, Gazette, Aust. Math. Soc.  32 (2005), 263-266.

This is the link. (Thanks to lhf for pointing out.)



