Convergence of $\int_0^1 \frac{\ln(1-x)\sqrt{x-x^2}}{\sin(\pi x)} \, dx$ I have difficulties with convergence of this integral:
$$\int_0^1 \frac{\ln(1-x) \sqrt{x-x^2}}{\sin(\pi x)} \, dx$$
I found similar problem here Covergence of integral but I don't get the solution from there.

I split the interval on two intervals: $(0,\alpha]$ and $[\alpha,1)$. The integral on the first interval convergs following a limit comparison test with $\frac{x^{3/2}}{\pi x}$.

But I am struggling with the second interval. Can somebody show me what to do on the second interval $[\alpha,1)$?
For $x$ close to $1$ the integrand

$$ \frac{\ln(1-x)\sqrt{x-x^3}}{\sin \pi x} \sim c\frac{\ln(1-x)}{\sqrt{x-1}} $$

which is an integrable function
For $x$ close to $1$ the integrand

$$ \frac{\ln(1-x)\sqrt{x-x^3}}{\sin \pi x} \sim c\frac{\ln(1-x)}{\sqrt{x-1}} $$

which is an integrable function