Integral of signum and a root

Question

I need to evaluate (or at least find an upper bound) of this type of integrals: $$ \int_0 ^{\frac{1}{2}} \text{sign}\left(\frac{1}{2} - x\right) \frac{1}{\left(\frac{1}{2}-x\right)^s} \text{d}x \,,$$ for $s \in (0,1)$ and where sign is the signum function. I know from Mathematica that the integral converges for a fixed value of $s$ but I need the general answer in terms of $s$. Thanks for any ideas on how to proceed!

Answer 1

Hint. By the change of variable $$ u=\frac12-x,\quad du=-dx, $$ observing that $\text{sign}\left(\frac{1}{2} - x\right)=1 $ for $0<x<\frac12$, one just gets $$ \int_0 ^{\frac{1}{2}} \text{sign}\left(\frac{1}{2} - x\right) \frac{1}{\left(\frac{1}{2}-x\right)^s} \text{d}x = \int_0 ^{\frac{1}{2}}\frac{du}{u^s},\quad s \in (0,1) . $$