quality:correction, structural:summarization, linguistic:paraphrasing
Generalized midpoint rule

A generalized midpoint rule formula, also known as the enhanced midpoint integration, is given by ∫ 0 1 f (x) d x = 2 ∑ m = 1 M ∑ n = 0 ∞ 1 (2 M) 2 n + 1 (2 n + 1) ! f (2 n) (x) | x = m − 1 / 2 M, {\displaystyle \int _{0}^{1}f(x)\,dx=2\sum _{m=1}^{M}{\sum _{n=0}^{\infty }{{\frac {1}{{\left(2M\right)^{2n+1}}\left({2n+1}\right)!}}{{\left.f^{(2n)}(x)\right|}_{x={\frac {m-1/2}{M}}}}}}\,\,,} where f (2 n) {\displaystyle f^{(2n)}} denotes even derivative.

For a function g (t) {\displaystyle g(t)} defined over interval (a, b) {\displaystyle (a,b)}, its integral is ∫ a b g (t) d t = ∫ 0 b − a g (τ + a) d τ = (b − a) ∫ 0 1 g ((b − a) x + a) d x. {\displaystyle \int _{a}^{b}g(t)\,dt=\int _{0}^{b-a}g(\tau +a)\,d\tau =(b-a)\int _{0}^{1}g((b-a)x+a)\,dx.} Therefore, we can apply this generalized midpoint integration formula by assuming that f (x) = (b − a) g ((b − a) x + a) {\displaystyle f(x)=(b-a)\,g((b-a)x+a)}. This formula is particularly efficient for the numerical integration when the integrand f (x) {\displaystyle f(x)} is a highly oscillating function.