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

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The discussion centers on a user encountering different results for the same function's indefinite integrals. The confusion arises from the absolute value properties, specifically that |x-1| and |1-x| are equivalent, leading to identical integrals despite appearances. A clarification is made that the negative sign in the final expression is consistent and should not be disregarded. The user is reassured that both integrals are indeed the same, and the negative sign does not alter their equivalence. Understanding the properties of absolute values resolves the discrepancy in results.
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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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