Does $\int^1_0\frac{(1-s)^\alpha}{s^\beta} \operatorname d \!s$ converge?

Question

I want to ask about the finitess of the following integral

\begin{equation} \int^1_0\frac{(1-s)^\alpha}{s^\beta}ds \end{equation}

when $\alpha>\beta>1$. This integral is very similar to the Beta function, aside from the negative power $-\beta$ here. Can anyone confirm if there really exists some big $\alpha$ such that the integral is finite? Thank you.

Answer 1

The integral is finite if and only if $$\beta<1\quad\text{and}\quad -\alpha<1\iff \beta<1\quad\text{and}\quad \alpha>-1$$