Confused about integral of f'(x)/f(x)

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Discussion Overview

The discussion revolves around the integration of the function f'(x)/f(x) and the implications of integrating equivalent fractions. Participants explore the confusion arising from obtaining different integral results for seemingly equivalent expressions.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about integrating two equivalent fractions, specifically 2x/x^2 and 6x/3x^2, and questions why their integrals yield different results.
  • Another participant points out that the difference in results is due to the constant of integration, noting that ln(3x^2) can be expressed as ln(x^2) + ln(3).
  • A later reply confirms the importance of the constant of integration in resolving the apparent discrepancy.

Areas of Agreement / Disagreement

Participants generally agree that the difference in the integral results is related to the constant of integration, but the initial confusion about the integration process remains unresolved.

Contextual Notes

The discussion does not fully address the implications of the constant of integration in the context of the original question, leaving some assumptions about the integration process unexamined.

Lucy Yeats
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A fraction obviously stays the same if you multiply top and bottom by the same constant. If you integrate two equal functions, surely you should get the same answer? By applying the rule in the title, you can get different answers. I'm very confused.

Example: 2x/x^2=6x/3x^2
The integral of 2x/x^2 is ln(x^2).
The integral of 6x/3x^2 is ln(3x^2)

How come these aren't equal? Is it to do with the +c?
 
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Remember that \ln (ab) = \ln a + \ln b. Hence \ln(3x^2)=\ln(x^2)+\ln(3), so the two expressions just differ by a constant.
 
Ah, okay. Thanks!
 
Lucy Yeats said:
A fraction obviously stays the same if you multiply top and bottom by the same constant. If you integrate two equal functions, surely you should get the same answer? By applying the rule in the title, you can get different answers. I'm very confused.

Example: 2x/x^2=6x/3x^2
The integral of 2x/x^2 is ln(x^2).
The integral of 6x/3x^2 is ln(3x^2)

How come these aren't equal? Is it to do with the +c?

Yes, it is about the plus C.
ln(3x^2)=ln(3)+ln(x^2)
 

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