Improper integral: is it convergent?

Question

Is this integral finite? $$\int_s^t \frac{dx}{x^{1/2} - s^{1/2}}$$ where $s,t \in (0,\infty)$. More generally, I have the following integral $$\int_s^t \frac{dx}{\left(x^{2\beta}-s^{2\beta}\right)^{k/2}}$$ where $k\geq 1$ is a fixed integer and $\beta \in (0,1/k)$. I wonder if the integral is finite. If not, can one adjust $\beta$ even more to make it finite? Thanks!

Answer 1

Hint: $$\int_{s}^t \frac{1}{x^{1/2}-s^{1/2}} dx= \int_{s}^t \frac{x^{1/2}+s^{1/2}}{x-s}dx, $$ now substitute $y=x-s$ you get $$\int_0^{t-s} \frac{x^{1/2}+s^{1/2}}{y}dy\geq \int_0^{t-s} \frac{2s^{1/2}}{y}dy \ldots$$