Why are my integrals giving different results for the same function?

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    Integrals
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Discussion Overview

The discussion revolves around the discrepancies encountered when integrating a function that appears to yield different results. Participants are examining the implications of absolute values in the integrals and the treatment of signs in the final expressions.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant expresses confusion over obtaining two different indefinite integrals for the same function, questioning if there is an error in their approach.
  • Another participant asserts that the two integrals are identical due to the properties of absolute values, specifically that |x-1| equals |1-x|.
  • A third participant reinforces the idea that the absolute values lead to the same result, emphasizing that both expressions are equivalent.
  • One participant raises a concern about the appearance of a negative sign in the final expression, questioning how it can be disregarded when variables can take on both positive and negative values.
  • Another participant counters that the final expressions indeed both contain a negative sign and argues that there is no need to eliminate it.

Areas of Agreement / Disagreement

Participants disagree on the treatment of the negative sign in the final expressions and whether it can be disregarded. While some assert the integrals are equivalent, the discussion remains unresolved regarding the implications of the negative sign.

Contextual Notes

The discussion highlights potential limitations in understanding the treatment of absolute values and the implications of variable signs in integration, but does not resolve these issues.

MartinV05
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I've been solving this exercise and I came to a point when one function can get two different integrals:
integral.jpg

Am I doing something wrong? Because both functions are the same, and the integrals (indefinite) are really different. This is a huge problem, because this is almost the final step of an exercise and when I exchange the current variable (x) with the previously defined function for it, the solution is VERY different.
**There should be a "-" in front of the last line of equation in the picture.
 
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No, the two integrals are NOT "really different"- they are identical. |x- 1|= |1- x| so, of course, -ln(|x-1|)+ C= -ln(|1- x|)+ C.
 
Since you are taking the absolute value, |1-x| and |x-1| are the same.
 
In the final expression a "-" appears, but I don't see how we can just make it go away (turn positive) when we are working with variables. The variable can be +-∞.
 
The final expressions both have a "-". There is no need to make it go away.
 

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