It seems the "trick" is that ArcTan's derivative is a simple inverse-polynomial? Once you have this, it seems that 22/7 could be replaced by many other fractions by adjusting the degrees and coeffs of the integrand (of course result wouldn't always be positive!)
This is an elegant little proof that 22/7 is bigger than pi. It's quite remarkable and unexpected (to me) that such a proof should be possible. One naturally wonders if something deeper is going on (and how the proof was found).https://twitter.com/johncarlosbaez/status/1083765566790283264 …
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With some cleverness, one should be able to change the function being integrated to get proofs that other constants are bigger than pi, so it's definitely interesting, but I'm not sure it's terribly related to 22/7.
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Quite accessible paper showing how this generalizes, with proof that 355/113 > pi http://www.austms.org.au/Gazette/2005/Sep05/Lucas.pdf …
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The \pi comes from the denominator, and you can sub in various other even exponents in the numerator to get other such approximations. it's nice, & also I am curious what is the particularly unexpected part for you?
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