calculus - Proving that $int^1_0 frac{1}{sqrt{ln(frac{1}{x})}}dx$ converges?

How do you prove that $\int^1_0 \frac{1}{\sqrt{\ln(\frac{1}{x})}}dx$ converges? I've tried more or less everything I can think of and still can't get the answer. Any hints will be appreciated!

Answer

HINT

We have that

$$\int^1_0 \frac{1}{\sqrt{\ln(\frac{1}{x})}}dx =\int_1^\infty \frac{1}{x^2\sqrt{\ln x}}dx =\int_1^2 \frac{1}{x^2\sqrt{\ln x}}dx+\int_2^\infty \frac{1}{x^2\sqrt{\ln x}}dx$$

and then refer to limit comparison test.