[(|f(x)|)^(-1)]*integral(|f(x)|dx) = [(f(x))^(-1)]*integral(f(x)dx)?

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In summary, the conversation discusses whether the statement (|f(x)|)^(-1)*integral(|f(x)|dx) = (f(x))^(-1)*integral(f(x)dx) is true, and if so, how. It is clarified that the numerator of the fraction is a number, while the denominator is a function. The question is then posed whether the statement holds for all values of x and whether the integration is definite or indefinite. Ultimately, it is determined that the statement is not true, as a counter-example is provided.
  • #1
coverband
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I heard [(|f(x)|)^(-1)]*integral(|f(x)|dx) = [(f(x))^(-1)]*integral(f(x)dx)

(i.e. when dividing the integral of the absolute value of a function by the absolute value of the function, the absolute value signs cancel out)

Is this true

If so, how so?
 
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  • #2
Are these antiderivatives? If so, the presence of unknown constants are going to blow this whole thing out of the water. Regardless, let's pick an example

f(x) = x

[tex] \frac{ \int_{-a}^{a}|x|dx }{ |a| } = |a| [/tex]
[tex]\frac{ \int_{-a}^{a} xdx } { a } = 0[/tex]

Like that?
 
  • #3
Basically is
(int|x|dx)/(|x|) = (intxdx)/( x )

and if so how?
 
Last edited:
  • #4
First let's clear up the notation.
The numerator
[tex]\int f(x) \, dx[/tex]
is a number, it does not depend on x. The denominator f(x) does.
Do you then claim that
[tex]\frac{ \int |f(y)| \, dy}{|x|} = \frac{\int f(y) \, dy}{x}[/tex]
for all x?

Secondly, is the integration definite (e.g. implied over the whole domain) or indefinite (i.e. you need an arbitrary integration constant to be added)?

Finally, Office_Shredder has given a counter-example to what was supposedly your statement, so to answer your question: "No, basically it is not".
 
  • #5
CompuChip said:
First let's clear up the notation.
The numerator
[tex]\int f(x) \, dx[/tex]
is a number, it does not depend on x. The denominator f(x) does.
Do you then claim that
[tex]\frac{ \int |f(y)| \, dy}{|x|} = \frac{\int f(y) \, dy}{x}[/tex]
for all x?

Secondly, is the integration definite (e.g. implied over the whole domain) or indefinite (i.e. you need an arbitrary integration constant to be added)?

Finally, Office_Shredder has given a counter-example to what was supposedly your statement, so to answer your question: "No, basically it is not".


first of all, why should the numerator be a "number" if the integral is indefinite, then it gives a function! only if the num was definite, would it be a number
 

Related to [(|f(x)|)^(-1)]*integral(|f(x)|dx) = [(f(x))^(-1)]*integral(f(x)dx)?

What is the meaning of [(|f(x)|)^(-1)]*integral(|f(x)|dx) = [(f(x))^(-1)]*integral(f(x)dx)?

This equation is a representation of the inverse property of integrals, which states that taking the integral of the absolute value of a function and then taking the absolute value of the integral will result in the same value as taking the integral of the function itself and then taking the inverse of that value.

How is this equation useful in scientific research?

This equation is useful in a variety of scientific fields, such as physics, biology, and economics. It allows researchers to calculate the area under a curve or the total value of a quantity over a given interval, even if the function has both positive and negative values. This can help in analyzing data and making predictions.

Can this equation be applied to any type of function?

Yes, this equation can be applied to any type of continuous function. It is important to note that the function must be integrable, meaning that it can be represented as the area under a curve. This excludes functions that have infinite or undefined values.

What are the limitations of this equation?

One limitation of this equation is that it only applies to continuous functions. It also assumes that the function is integrable, which may not always be the case. Additionally, this equation does not account for functions that have discontinuities or undefined points.

Can this equation be used to solve real-world problems?

Yes, this equation can be used to solve a variety of real-world problems, such as calculating the total cost of a product over a given time period or determining the average speed of an object over a distance. It can also be used in optimization problems to find the maximum or minimum value of a function.

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