The number pi and its excellent rational approximation

The number pi is approximately equal to 355/113. Hence the continued fraction for pi includes some very large numbers at the beginning, e.g., 3; 7, 15, 1, 292, ...

The site in MathOverflow below shows the explanation for why pi is slighty smaller than 355/113 by using an appropriate integral.

https://mathoverflow.net/questions/425468/integral-representation-of-frac355113-pi

355/113-pi=integral of (x⁸ )(1-x)⁸ (25+816x² )/(3164(1+x² )) from x=0 to x=1.

Main question: Is the integral a coincidence?

Any good reason would be appreciated.