Median theorem to prove the integral equation? 50

No, the title does not say that it is inducible in (a, b).

The integral median theorem is a mathematical law. It is divided into the integral first median value theorem and the integral second median value theorem, each of which contains two formulas. Among them, the integral second median value theorem also contains three commonly used inferences.

The integral median value theorem reveals a method of integrating into function values, or integrating complex functions into the integration of simple functions, which is the basic theorem and important means of mathematical analysis, and is widely used in finding limits, determining certain property points, and estimating integral values.

The geometric significance of this theorem is as follows: The integral median theorem plays an important role in the application of the integral sign by removing the integral sign, or making the complex integrand function into a relatively simple integrand, thus simplifying the problem. Therefore, when proving that the relevant problem contains an equation or inequality that contains the integral of a function, or that the conclusion to be proved contains a definite integral, or that the limit formula sought contains a definite integral, it is generally necessary to consider using the integral median theorem, removing the integral sign, or simplifying the integrand.

Integral inequality refers to the inequality containing more than two integrals, when the integral interval is the same, first merge different integrals on the same integral interval, according to the conditions satisfied by the integrand, flexibly apply the integral median value theorem, in order to achieve the purpose of proving the inequality to be true.

When proving definite integral inequalities, the integral median theorem is often used to remove the integral sign, and if the integrand is the product of two functions, the first or second median theorem of the integral can be considered. For the proof of some inequalities, the median value theorem of the original integral can only be used to obtain the conclusion of " ", or the inequality cannot be proved at all. By applying the improved median theorem of the integral, we can get "".

or a successful solution to the problem.

I think the condition for using the median theorem is that the function is continuous, but f(x)g(x) you can't be continuous, but you should first prove that the function is continuous.

Integral median theorem: The integral of f(x) on a to b is equal to (a-b)f(c), where c satisfies a If the function f(x) is continuous on the integral interval [a, b], then at least one point exists on [a, b], making the following equation true.

It is divided into the first median theorem of the integral and the second median theorem of the integral part.

The integral median theorem is a kind of mathematical law, which is divided into the first median theorem of the integral and the second median theorem of the integral, each of them contains two formulas, of which the second median theorem of the integral also contains three commonly used inferences, the median theorem of the integral reveals a method of integrating into a function value, or integrating a complex function into an integral of a simple function, which is the basic theorem and important means of mathematical analysis, and is widely used in finding the limit, determining the balance circle of certain property points, and estimating the integral value.

Therefore, when proving that the relevant problem contains an equation or inequality of a function's integral, or the conclusion to be proved contains a definite integral, or the limit formula sought contains a definite integral, it is generally necessary to consider using the integral median theorem, removing the integral sign, or simplifying the integrand.

Integral median theorem: The integral of f(x) rolling sun on a to b is equal to (a-b) f(c), where c satisfies a If the function f(x) is continuous on the integral interval [a, b], then at least one point exists on [a, b], so that the following equation holds.

where (a b).

The integral median value theorem reveals a method of integrating into function values, or integrating complex functions into simple functions, which is the basic theorem and important means of mathematical analysis, and is widely used in finding limits, determining certain property points, and estimating integral values.

<> integral median theorem is divided into the integral first median theorem and the integral second median theorem, each of which contains two formulas. Its degenerative state refers to the existence of a moment in the process of change to make the area of the two figures equal.

The integral median value theorem reveals a method of integrating into function values, or integrating complex functions into the integration of simple functions, which is the basic theorem and important means of mathematical analysis, and is widely used in finding limits, determining certain property points, and estimating integral values.

1. If f and g are both continuous on [a,b], and g does not change on [a,b], then there is at least one point c that belongs to [a,b], such that f multiplied by g on [a,b] is equal to f(c) multiplied by g on [a,b].

2. Let the function f be integrable on [a,b]. If g is a monotonic function, then there is a point c that belongs to [a,b] such that the integral of (f multiplied by g) is equal to g(a) multiplied by (f integral on [a,c]) plus g(b) multiplied by (f integral on [c,b]).

1. Find the limit.

In the calculation of the limit of the function, if there is a definite integral formula, you can often use the relevant knowledge of the definite integral, such as the integral median value theorem, etc., and use some functions with integral formulas to apply the integral problem, which often requires the determination of certain properties of the point, and sometimes the use of the integral median value theorem can make the problem easy to solve.

2. Use estimation.

In most integrals, it is rare to find the original function of the integrand and evaluate the integral, and when the integrand "does not integrate" or the original function is complex, various methods can be used to estimate the integral. For the integrand of the product type, the part that changes slowly or the part that is difficult to integrate is estimated, and the integrable part is integrated. The integral median theorem and various inequalities are commonly used methods, 3. Inequality proof.

Integral inequality refers to the inequality containing more than two integrals, when the integral interval is the same, first merge different integrals on the same integral interval, according to the conditions satisfied by the integrand, flexibly apply the integral median value theorem, in order to achieve the purpose of proving the inequality to be true.

When proving definite integral inequalities, the integral median theorem is often used to remove the integral sign, and if the integrand is the product of two functions, the first or second median theorem of the integral can be considered. For the proof of some inequalities, the median value theorem of the original integral can only be used to obtain the conclusion of " ", or the inequality cannot be proved at all. By applying the improved integral median theorem, we can get a ">" conclusion or solve a successful problem.

Integral median value theorem: It is divided into the first median value theorem of the integral and the second median theorem of the integral, each of which contains two formulas. The degraded state refers to the existence of a moment in the process of change where the area of the two figures is equal.

If the function f(x) is continuous on the closed interval [a, b], then there is at least one point on the integral interval [a, b], so that the lower limit a is established on the upper selling limit b f(x)dx=f( )b-a)( a b).

Proof: Since f(x) is a continuous function on the closed interval [a,b], let the maximum and minimum values of f(x) be m and m respectively, so m f(x) m integrates the above equation in the interval [a,b] at the same time, and the integral median theorem m(b-a) The lower limit a upper limit bf(x)dx m(b-a) is the upper limit of m lower bound a bf(x)dx (b-a) mBecause m f(x) m is a continuous function, from the intermediate value theorem, there must be a point such that the upper limit of lower bound a bf(x)dx ( b-a)=f( ), i.e. lower limit a, upper limit bf(x)dx=f( )b-a)<>