# An integral inequality

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In summary, an integral inequality is a mathematical statement that compares the value of an integral to another value. It is unique because it deals with continuous functions and has a wide range of applications in various fields. These inequalities can be solved using different techniques but may have limitations in terms of accuracy and complexity.

## Homework Statement

suppose f(x) is monotonely decreasing and positive on [2,+∞),
here "∫ "means integrating on the interval [2,x]

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## The Attempt at a Solution

Maybe the second mean value thereom of integral is helpful.

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Have you tried anything? In particular, have you selected some simple monotonically decreasing function, such as $f(x)= \frac{1}{x}$ and calculated those two values?

In fact, yes!
But what I'm really eager to know is how to prove the conclusion.
Maybe when the x is large enough, [∫f(t)dt]^2 is larger.

## 1. What is an integral inequality?

An integral inequality is a mathematical statement that compares the value of an integral (a mathematical concept used to find the area under a curve) to another value. It is typically used to establish relationships between different functions or to prove mathematical theorems.

## 2. How is an integral inequality different from other types of inequalities?

An integral inequality is unique because it deals with the concept of an integral, which is a continuous function, rather than a discrete function. This means that the value of the integral is dependent on the entire shape of the function, rather than just a few specific points.

## 3. What are some common applications of integral inequalities?

Integral inequalities are used in a variety of fields, including physics, engineering, and economics. They can be used to model real-world situations, solve optimization problems, and prove mathematical theorems.

## 4. How are integral inequalities solved?

Integral inequalities can be solved using a variety of techniques, including algebraic manipulation, substitution, and integration by parts. The specific approach will depend on the structure of the inequality and the desired outcome.

## 5. Are there any limitations to using integral inequalities?

While integral inequalities are a powerful tool in mathematics and science, they do have some limitations. For example, they may not always provide the most precise solution to a problem and may require certain assumptions or conditions to be valid. Additionally, solving integral inequalities can be complex and time-consuming.