积分中值定理的推广

The extension of intermediate value theorem of integral

将Riemann积分中值定理中函数f(x)所满足的条件加以改进,得到如下积分中值定理:若函数f(x)是 闭区间[a,b]上有原函数的可积函数,函数g(x)在 [a,b]上可积且不变号,则存在ξ∈(a,b),使得 ∫ a b f(x)g(x)dx=f(ξ)∫a b g(x)dx.

To improve the conditions that meet the needs of the function f(x) in the intermediate value theorem of integral, the following intermediate value Riemann theorem is got: If function f( x ) is the integrable function of the o riginal function in the closed interval [ a, b ], and function g(x) in [a, b] is integrable and the sign is not changeable, then ξ∈(a,b), and the result is ∫a b f(x)g(x)dx = f(ξ)∫ a b g(x)dx.