How to Integrate ln(sin x) using Euler's Identity?

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SUMMARY

The integration of ln(sin x) can be approached using Euler's Identity, specifically by rewriting sin(x) in terms of exponential functions. The suggested method involves factoring out an exponential, expressing the logarithm as a sum of simpler logarithms, and performing a substitution. However, the resulting integral, ln(u)/(1 - u), remains unsolved and may require further techniques for resolution.

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  • Understanding of Euler's Identity in complex analysis
  • Familiarity with integration techniques and substitutions
  • Knowledge of logarithmic properties and their applications in calculus
  • Basic proficiency in handling integrals involving transcendental functions
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  • Explore the use of substitution methods in calculus
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Can somebody show me how to integrate this:?

[tex] \int ln (sin x) dx[/tex]

thanks
 
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Do you have any reason to believe that its anti-dervivative is an elementary function?
 
I tried a sloppy method that got me close:
1) use Euler's ID for sin(x)
2) factor out an exponential
3) write the ln as a sum of simpler ln's
4) do a substitution for one of the ln's

This got me to an integral I didn't know how to solve, but that is probably simpler in principle. The integrand is:
ln(u)/(1 - u)
 

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