The book "Build supercomputers with Raspberry PI" (C. Morrison) calculates a numerical approximation of PI using the following integral:
$${\pi} = \int_{0}^1 \frac{4}{1+x^2}\ dx $$
Who is credited with this derivation? I could not find it among the named formulas (Euler, Ramanujan, etc) for calculating PI.
On
$$\frac{1}{1+x^2}=1-x^2+x^4-x^6+x^8-...., x^2 \le 1$$ Integrate w,r, t, x you get the famos Newton-Gregory series $$\tan^{-1} x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+,,,,,$$ Which has been helping in computing the value of $\pi$ $(x=1)$correct to several thousand decimal places to if any string pf numbers repeats to decide if $\pi$ is a rational number. But $\pi$ every time turns out to be irrational.
A lot of things in mathematics are nameless, and this formula is one of them. As discussed in comments, the antiderivative of the integrand is simply $4\tan^{-1}x$, and substituting $x=1$ and simplifying gives $\pi$.