If the functions f(x) and g(x) are continuous on the closed interval (a, b), and? G(x) has the same sign on (a, b), so there is at least one point on the integration interval (a, b). , thus establishing the following formula:

The second theorem

First, if the function? F(x), right? G(x) is integrable on the closed interval (a, b), and? If f(x) is a monotone function, there is at least one point on the integral interval (a, b), so the following formula holds:

Second, if the function? F(x), right? G(x) is integrable on the closed interval (a, b), and f (x) >; =0 is a monotonically decreasing function, so there is at least one point on the integral interval [a, b], so the following formula holds:

Third, if the function? F(x), right? G(x) is integrable on the closed interval (a, b), and f (x) >; =0 is a monotonically increasing function, so in the integer interval [a, b]? The following formula holds at least on one point:

Extended data

The integral mean value theorem plays an important role in application, which can remove the integral sign or make the complex integrand function become a relatively simple integrand function, thus simplifying the problem. Therefore, when proving an equation or inequality containing functional integral in related problems, or when the conclusion to be proved contains definite integral, or when the limit formula to be proved contains definite integral, we should generally consider using the mean value theorem of integral, removing the integral symbol or simplifying the integrand function.

The mean value theorem of integrals reveals a method of transforming integrals into function values or complex functions into simple functions. It is a basic theorem and an important means of mathematical analysis, which is widely used in finding limits, judging some property points, estimating integral values and so on.

Refer to Baidu Encyclopedia-Integral Mean Value Theorem