Question regarding integrating (int(1/u))

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SUMMARY

The integral of the function \(\int \frac{1}{u} du\) is definitively expressed as \(\ln |u|\). This is because the absolute value is necessary to ensure the logarithm remains defined for both positive and negative values of \(u\). While \(\ln(u)\) may appear acceptable in certain contexts, it can lead to undefined results if \(u\) is negative. The discussion clarifies that using absolute values is crucial for maintaining mathematical rigor in integration, particularly when applying the u-substitution method.

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  • Understanding of basic calculus concepts, specifically integration.
  • Familiarity with the u-substitution method in integration.
  • Knowledge of logarithmic functions and their properties.
  • Ability to differentiate between positive and negative values in mathematical expressions.
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  • Study the properties of logarithmic functions, focusing on the significance of absolute values.
  • Practice more examples of u-substitution in integration to solidify understanding.
  • Explore the implications of undefined values in logarithmic functions when integrating.
  • Review calculus textbooks or online resources that cover integration techniques and their applications.
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Students learning calculus, particularly those struggling with integration techniques and the use of logarithmic functions. This discussion is also beneficial for educators seeking to clarify common misconceptions in teaching integration methods.

Linday12
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Homework Statement


I'm confused about integrating something like [tex]\int[/tex][tex]\frac{1}{u}[/tex]
Sometimes the answer seems to be ln|u| and sometimes just ln(u), and I wasn't sure why it is different from problem to problem. (after subbing u back in; I'm using the u-sub method)

It looks like the answer should be very straight forward as I can't find an explanation, so I'm not seeing something. My only guess would be that the absolute value symbols are used only if a negative value of x could give a negative value for u when subbing back in, making the ln(blah) undefined (for my course anyways). Is this about right? Thanks.
 
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Don't worry about it. The integral should be ln(|u|) because deriving that gives you 1/u for both u>0 and u<0. Sometimes, writing ln(u) works because (1) it's convenient and (2) it often works out in the end even if u is negative and the absolute value sign is ignored.
 

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