Why is this proof wrong?

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In summary: I don't understand.In summary, the 0 = 1 proof is flawed because it does not use integration by parts.
  • #1
Another one of those 0 = 1 proofs.

If you dislike them, please navigate away to cats doing things with captions. Otherwise stay tuned.

Tell me what's wrong (obviously I know what's wrong but I just want to put it out there for people to mull over).

Note, you need to know basic calculus.

[tex]

\int \frac{1}{x} dx = \int \frac{1}{x} dx
[/tex]

[tex]
u = \frac{1}{x}
[/tex]

[tex]
dv = dx
[/tex]

[tex]
du = \frac{-1}{x^2} dx
[/tex]

[tex]
v = x
[/tex]

[tex]
\int \frac{1}{x} dx = u * v - \int v du
[/tex]

[tex]
\int \frac{1}{x} dx = \frac{1}{x} * x - \int x * \frac{-1}{x^2} dx
[/tex]

[tex]
\int \frac{1}{x} dx = 1 - \int \frac{-1}{x} dx
[/tex]

[tex]
\int \frac{1}{x} dx = 1 + \int \frac{1}{x} dx
[/tex]

[tex]
0 = 1
[/tex]
 
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  • #2
Right off the bat, [tex]u = \frac{{dv}}{x} = dx[/tex] makes no sense. The second line makes no sense either... am i completely missing something?
 
  • #3
Sorry those are supposed to be separate. I will edit that in.

This is what it should look like

[tex]
u = \frac{1}{x}
[/tex]

[tex]
dv = dx
[/tex]

I have also added an extra step just to show that I am going to be using integration by parts to do the "proof"
 
  • #4
[tex]\int \frac{1}{x} dx = 1 - \int \frac{-1}{x} dx[/tex] isn't valid. Integration by part goes like:

[tex]\int\limits_a^b {udv} = [uv]_a^b - \int\limits_a^b {vdu} [/tex]

The term you think is 1 is actually 0.
 
  • #5
We are dealing with antiderivatives. So, given F(x) and G(x) that are antiderivatives of 1/x, it is true that F(x) = G(x) + C, for some constant C. For example, the second to last line you have can be written as

ln(x) + C = 1 + ln(x) + D for some constants C, and D. We do not then conclude that 0 = 1.

The integral of 1/x dx is a FAMILY of functions that all differ by a constant.
 
  • #6
I only got to the end of the second sentence
 
  • #7
Russell has the right answer. To be strictly correct, the rule for integration by parts is better written as

[tex]\int u dv = u*v - \int v du + C[/tex]

We drop the arbitrary constant because it is implied by the very use of indefinite integrals, or antiderivatives. The inverse of the derivative is not unique.
 
  • #8
Russell and D_H are correct, the constant term is missing. Gj guys :)
 
  • #9
protonchain said:
Russell and D_H are correct, the constant term is missing. Gj guys :)

and Pengwuino too. You either put in constants of integration, or use definite integrals. Either way resolves the error.
 
  • #10
Right, but since I defined the problem in the start as indefinite integrals, I was looking for the answer that was related to indefinite integrals. Anyways. It's just a simple problem
 

Question 1: Why is this proof wrong?

There are many reasons why a proof may be incorrect. It could be due to a logical error, an incorrect assumption, a mistake in calculations, or an incomplete analysis.

Question 2: How do I know if a proof is wrong?

A proof is considered wrong if it violates one or more of the accepted rules of logic and mathematics. This can be identified through careful examination of the steps, assumptions, and conclusions made in the proof.

Question 3: Can a wrong proof still lead to a correct conclusion?

Yes, it is possible for a proof to have incorrect steps or assumptions, but still arrive at a correct conclusion. However, this does not make the proof itself valid or reliable.

Question 4: What should I do if I find an error in a proof?

If you find an error in a proof, you should carefully analyze and identify the mistake, and then communicate your findings to the person who presented the proof. This can help improve the overall understanding and accuracy of the proof.

Question 5: How can I avoid making mistakes in my own proofs?

To avoid making mistakes in your proofs, it is important to carefully plan and organize your steps, check your assumptions, and double check your calculations. It can also be helpful to have someone else review your proof for any potential errors.

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