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I am looking for proof of generalized open & closed Newton-Cotes formulas. I couldn't find any reference which properly proves both theorems. Most books just state the theorem and do not provide any proof.

Suppose that $\sum_{i=0}^{n} a_{i} f\left(x_{i}\right)$ denotes the $(n+1)$ -point open Newton-Cotes formula with $x_{-1}=a, x_{n+1}=b,$ and $h=(b-a) /(n+2) .$ There exists $\xi \in(a, b)$ for which $$ \int_{a}^{b} f(x) d x=\sum_{i=0}^{n} a_{i} f\left(x_{i}\right)+\frac{h^{n+3} f^{(n+2)}(\xi)}{(n+2) !} \int_{-1}^{n+1} t^{2}(t-1) \cdots(t-n) d t $$ if $n$ is even and $f \in C^{n+2}[a, b],$ and $$ \int_{a}^{b} f(x) d x=\sum_{i=0}^{n} a_{i} f\left(x_{i}\right)+\frac{h^{n+2} f^{(n+1)}(\xi)}{(n+1) !} \int_{-1}^{n+1} t(t-1) \cdots(t-n) d t $$ $ \text { if } n \text { is odd and } f \in C^{n+1}[a, b] \text { . } $

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You can find the proof in "Isaacson, E. and Keller. Analysis of Numerical Methods" chapter 7.

NOTE: To preview the book, one must have to borrow the book and to do that one must have an login account on Archive. You can create a for account for free.

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  • $\begingroup$ My question is not answered in that book. No proof has been provided. $\endgroup$ – FreeMind Jan 19 at 20:58
  • $\begingroup$ @ FreeMind. In the version of 1994 of this book, you can find the proof in page 310 for the even case and in page 311-312 for the odd case. The authors prove the theorems before state it. $\endgroup$ – Houssam Jan 22 at 18:27
  • $\begingroup$ @HiterDean thanks for the information $\endgroup$ – Houssam Jan 22 at 18:28

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