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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.

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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?

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  • $\begingroup$ Yes, thank you! $\endgroup$
    – Ellie
    May 14 at 5:32

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