一个积分不等式的多种证明方法

Multiple Proofs of an Integral Inequality

关于定积分有许多著名且重要的不等式,几乎每个不等式都有很多种证法,可以用定义,可积条件,定积分性质,定理,借助已知不等式,计算定积分以及借助各种技巧来证明。本文采用多种方法证明积分不等式∫abxf(x)dx≥a+b/2∫abf(x)dx(其中,f(x)在区间[a,b]上连续,且单调增加),借此归纳积分不等式证明方法。

There are many famous and important about definite integral inequality, there are many kinds of almost every inequality proof, can be used to define,integrable condition, the definite integral property, theorem, with the aid of known inequality,calculation of definite integral, and use a variety of techniques to prove. In this paper, the integral inequality ∫abxf（x）dx≥a＋b/2∫abf（x）dx （f（x） is continuous on the interval [a, b], and monotonically increasing） .Thisinduces the proof of the integral inequality.