Midpoint Derivative-Based Closed Newton-Cotes Quadrature

Table 1

Precision, the orders and the error terms for midpoint derivative-based closed Newton-Cotes quadrature.

𝑛	Precision	Order	Error terms
Trapezoidal rule (𝑛=1)	3	5	−(𝑏−𝑎)^5𝑓480^(4)(𝜉)
Simpson’s rule (𝑛=2)	5	7	−(𝑏−𝑎)^7𝑓241920^(6)(𝜉)
Simpson’s 3/8 rule (𝑛=3)	5	7	−23(𝑏−𝑎)^7𝑓9797760^(6)(𝜉)
Bool’s rule (𝑛=4)	7	9	−17(𝑏−𝑎)^945⋅2^11𝑓⋅8!^(8)(𝜉)