Absolute values in standard integrals

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SUMMARY

The discussion centers on the use of absolute values in the solutions of standard integrals, specifically in the integral of the cotangent function, \(\int dx \cot{x}\), which is stated to equal \(\log |\sin{x}|\). Participants express confusion regarding the necessity of absolute values, arguing that \(\log (\sin{x})\) should suffice. The consensus acknowledges that the absolute value is included to ensure the function remains defined across its domain, particularly since \(\sin{x}\) can take negative values, thus making \(\log(-x)\) undefined.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly integration.
  • Familiarity with trigonometric functions and their properties.
  • Knowledge of logarithmic functions and their domains.
  • Awareness of the implications of absolute values in mathematical expressions.
NEXT STEPS
  • Study the properties of trigonometric integrals, focusing on \(\int \cot{x} \, dx\).
  • Explore the implications of absolute values in logarithmic functions.
  • Learn about the continuity and differentiability of functions involving absolute values.
  • Investigate common pitfalls in calculus related to the domains of functions.
USEFUL FOR

Students of calculus, mathematics educators, and anyone interested in the nuances of integral calculus and the behavior of trigonometric functions.

NanakiXIII
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In a lot of compilations of standard integrals (my Calculus book does this, Wikipedia does this), a lot of the integrals of trigonometric functions have an absolute value in their solution which seems out of place to me. For example, take the integral

\int dx \cot{x}.

My Calculus book says this equals \log |\sin{x}|. Now, I could be mistaken, but it seems to me that \log (\sin{x}) would do just fine and in fact, since |\sin{x}| isn't a smooth function, I wouldn't expect to find it as a solution.

My first guess is that the | | are for some reason used without their meaning as absolute value. Is this the case? If so, what's the point of using them instead of regular brackets?
 
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log(-x) isn't defined.
 
You are right, of course. And the cotangent is not a bounded function, so the solution not being smooth is not so strange.
 

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