Is this function bounded? (Order of explosion of a function)

Question

I have the following function: $f:[0,1]\rightarrow \mathbb{R}$ defined as $$f(x)=\int_x^1 \frac{(y-x)^{-\alpha}}{y}dy, \quad x\in [0,1],$$ where $\alpha\in (0,1/2)$ is some fixed parameter.

To me it seems this function is continuous on $(0,1]$ but not at $0$. It diverges. I would like to know the order of such explosion. My guess is:

$$f(x) = x^{-\alpha} g(x),$$ where $$g(x) = x^{\alpha}\int_x^1 \frac{(y-x)^{-\alpha}}{y}dy.$$

If $g$ is bounded then I will have the order of explosion. The question is:

Is $g$ bounded? If not, how could I find out the order of explosion of $\lim_{x\to 0}f(x)$?

Thanks a lot!

Answer 1

Let $y=xt.$ Then

$$f(x) = \frac{1}{x^{a}}\int_1^{1/x}\frac{1}{(t-1)^at}\,dt.$$

Thus $f(x) \sim C/x^a,$ where

$$C = \int_1^\infty\frac{1}{(t-1)^at}\,dt.$$