\(\displaystyle \int\frac{cot(x)}{ln|sin(x)|}dx \ = \ \int\frac{cos(x)dx}{sin(x)[ln|sin(x)|]}\)
\(\displaystyle Let \ u \ = \ sin(x), \ then \ du \ = \ cos(x)dx\)
\(\displaystyle Ergo, \ \int\frac{du}{u[ln|u|]}\)
\(\displaystyle Now, \ let \ k \ = \ ln|u|, \ then \ dk \ = \ \frac{1}{u}du\)
\(\displaystyle Hence, \ we \ now \ have \ \int\frac{dk}{k} \ = \ \int k^{-1}dk \ = \ln|k|+C.\)
\(\displaystyle Resubstituting, \ we \ get \ ln[ln|u|]+C \ = \ ln[ln|sin(x)|] \ +C \ QED.\)
