Calculate integral for the function :

$\displaystyle

\begin{array}{l}

\ln (\sin (x)) \\

\ln (\cos (x)) \\

\end{array}$

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- Nov 24th 2009, 06:00 AMdhiabIntegrals : 2
**Calculate integral for the function :**

$\displaystyle

\begin{array}{l}

\ln (\sin (x)) \\

\ln (\cos (x)) \\

\end{array}$ - Nov 24th 2009, 07:21 AMJameson
- Nov 24th 2009, 07:26 AMDrexel28
- Nov 24th 2009, 07:32 AMJameson
- Nov 24th 2009, 07:40 AMDrexel28
Maybe we can make that a tad more apparent? Let $\displaystyle x=\arcsin(z)\implies dx=\frac{dz}{\sqrt{1-z^2}}$ so our integral becomes $\displaystyle \int\frac{\ln(z)}{\sqrt{1-z^2}\ln\left(\sqrt{1-z^2}\right)}$. Let $\displaystyle \sqrt{1-z^2}=\tau\implies \frac{-\tau}{\sqrt{1-\tau^2}}d\tau=dz$. So then our integral becomes $\displaystyle \frac{-1}{2}\int\frac{\ln\left(1-\tau^2\right)}{\sqrt{1-\tau^2}\ln\left(\tau\right)}d\tau$. That looks

*almost*doable. I am not feeling it right now. Maybe the OP can continue.