Is $f(x)=\frac{\sqrt{1-x}}{\ln{x}}$ on $[0,1]$ a Lebesgue-integrable function? 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? 

Hints: Near $0$ there is no problem. Near $1, \ln (x) \approx (x-1).$
