Could you help me with a comprehensive one? I have tried to solve it but I can't reach the result. The integral I am trying to evaluate is
$$\int_0^\pi\ln(\sin(\theta))\sin(\theta)d\theta$$and according to Wolfram alpha the result should be $\ln(4)-2$.
I already have tried to evaluate it using integration by parts but, when doing so, I find a term of $\cos(\theta)\ln(\sin(\theta)) $ which is evaluated from $0$ to $\pi$ and this is where I stay because those values make the result indeterminate. I have tried to make variable changes but I have not been able to find a suitable one that allows me to avoid indeterminate results. I would like you to provide me with a suggestion to avoid the indeterminacies.
Best Answer
You can do this: \begin{align*}&\ \ \ \ \int_0^\pi\ln(\sin(\theta))\sin(\theta)d\theta \\&= \frac{1}{2}\int_0^\pi\ln(1-\cos^2(\theta))\sin(\theta) d\theta= \\ &=\frac{1}{2}\left(\int_0^\pi\ln(1-\cos(\theta))d(1-\cos(\theta))-\int_0^\pi\ln(1+\cos(\theta))d(1+\cos(\theta))\right) \\ &=\frac{1}{2}\left((1-\cos(\theta))(\ln(1-\cos(\theta))-1)|_{0}^{\pi}-(1+\cos(\theta))(\ln(1+\cos(\theta))-1)|_{0}^{\pi}\right) \\ &=\frac{1}{2}(2\ln 2-2+2\ln 2 -2)\\&= \ln(4)-2 \end{align*}