## 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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