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A new generalization of the trapezoid formula for \(n\)-time differentiable mappings and applications. (English) Zbl 0974.26010

The following new generalization of the trapezoid formula is proved. Let \( f:[a,b]\rightarrow \mathbb{R}\) be a mapping such that the derivative \( f^{(n-1)}(n\geq 1)\) is absolutely continuous on \([a,b]\). Then \[ \int_{a}^{b}f(t) dt= \sum_{k=0}^{n-1}\frac{1}{(k+1)!} [(x-a)^{k+1}f^{(k)}(a) +(-1)^{k}(b-x)^{k+1}f^{(k)}(b)]+\frac{1}{n!}\int _{a}^{b}(x-t)^{n}f^{(n)}(t) dt, \] for all \(x\in [a,b].\) Some applications in numerical analysis and inequalities are also given.

MSC:

26D15 Inequalities for sums, series and integrals
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
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