For the definite integral of the integrand with a continuous derivative of order m+n ,the asymptotic expansion of the definite integral at the two points of the upper and lower bounds of the integral is given by multiple integration by parts.The remaining terms are similar to the remainder of the integral form of Taylor's formula,so this expansion can be regarded as a generalization of Taylor′s formula.Because of the different values of the integrand function at the upper and lower limits of the integral,the expansion will also change and have multiple forms.Through analysis and examples,it is found that the approximate calculation of this expansion formula is better than that of Taylor formula,and it is also fast and convenient in the proof of some integral inequalities.
关键词
定积分/渐进展开式/Taylor公式的积分型余项/分部积分
Key words
definite integral/asymptotic expansion/the integral remainder of Taylor's formula/integration by parts
维普
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