I've split the integral into:
$\int_0^1\frac{1-x^\alpha}{1-x}\mathrm dx=\int_0^{1/2}\frac{1-x^\alpha}{1-x}\mathrm dx+\int_{1/2}^1\frac{1-x^\alpha}{1-x}\mathrm dx$
I'm trying to find a suitable function in the form of $g(x)=\frac1{(x-1)^\text{?}}$ so I can use the LCT when $x\to 1$ but I can't figure out what needs to be the exponent. I need to somehow extract $(1-x)$ from $1-x^\alpha$.
How can this be done?