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?

**Answer**

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 that355/113 > \pi, Gazette, Aust. Math. Soc. 32 (2005), 263-266.This is the link. (Thanks to lhf for pointing out.)

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