# Proving convergence of $\int_0^1 |\frac{x^{1/2}-x^{-1/2}}{\ln{x}}|dx$

I'm trying to prove convergence of the following integral: $$\int_0^1 \left|\frac{x^{1/2}-x^{-1/2}}{\ln{x}}\right|dx.$$ First, I made a substitution $$x=1/t$$. Thus, I got the following integral: $$\int_{1}^{\infty} \left|\frac{\sqrt{\frac{1}{t}}-\sqrt{t}}{\ln{\frac{1}{t}}}\right|\frac{1}{t^2}dt.$$ The integrand function is bounded near $$1$$, and also approaches $$0$$ as $$t\to\infty$$, but I'm not sure how to properly prove that it converges.

I'll follow your idea to substitute $$x=1/t$$. That transforms the integral to $$\int_1^\infty\frac{t-1}{t^{5/2}\ln t}\,dt$$ Since $$\lim_{t\to 1^+}\frac{t-1}{\ln t}=1$$, the singularity at $$t=1$$ is removable, so we only have to worry about the convergence of $$\int_3^\infty\frac{t-1}{t^{5/2}\ln t}\,dt$$ For $$t\ge 3$$, we have $$0<\frac{t-1}{t^{5/2}\ln t}<\frac{t}{t^{5/2}\ln t}<\frac{t}{t^{5/2}}=\frac{1}{t^{3/2}}$$ Can you finish now?