I have to prove that $\displaystyle x\mapsto\frac{\sqrt{1-x}}{\ln{x}}$ is Lebesgue-integrable on $[0,1]$. So I try to bound $\displaystyle\left|\frac{\sqrt{1-x}}{\ln{x}}\right|$ with a Lebesgue-integrable function but I haven't succeed. Is there any idea?
2026-04-28 00:04:10.1777334650
Is $f(x)=\frac{\sqrt{1-x}}{\ln{x}}$ on $[0,1]$ a Lebesgue-integrable function?
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Hints: Near $0$ there is no problem. Near $1, \ln (x) \approx (x-1).$