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

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Integrating ln(sin x) involves using Euler's identity for sin(x) and manipulating the logarithm into simpler components. A proposed method includes factoring out an exponential and performing a substitution to simplify the integral. However, the resulting integral, ln(u)/(1 - u), remains complex and unsolved. The discussion highlights the challenge of determining whether the anti-derivative is an elementary function. Overall, the integration of ln(sin x) presents significant difficulties that require advanced techniques.
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Can somebody show me how to integrate this:?

<br /> \int ln (sin x) dx<br />

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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