the below integral is coming from M.Stein's complex analysis. chapter 3 exercise 9.
Show that $$\int_0^1\log(\sin \pi x)\text{d}x=-\log 2$$ use the contour shown in the above figure
Then I recall the definition of $\log(z)$,it must be defined on a simply connected domain $\Omega$ with $1\in \Omega$ and $0 \not\in \Omega$. In order to use the above contour to computer the integral. maybe consider $\int_{\gamma}\log(\sin \pi z)\text{dz}$,where $\gamma$ is the contour. but for $z=0$, there is no definition.
I think the author arrange the exercise here must expect the reader use residue formula to computer it. but I don't know how to. thanks very much
