Find $s$ so the integral converges

Question

I have the following integral:

$$ \int_0^1 \left( \frac{x}{(x-1)^2} \right) ^s dx$$

I need to find the values of $s$ for which the integral converges. I’ve tried using the comparison test but I only get $s<1$ (which is true) but does not help me to get to the answer which is $-1<s<\frac12$.

Does anybody have an idea of what else can i do?

Answer 1

Hint: The function is integrable iff $x^{s}$ is integrable near $0$ and $(x-1)^{-2s}$ is integrable near $1$. So the conditions are $s >-1$ and $s <\frac 1 2$.