I have improper integral $$\displaystyle \int_{1}^{2}\frac{dx}{\sqrt{(2-x)\ln(x)}}$$ and I need to find out if it converges/diverges. Now I don't if my thinking is correct:
$\displaystyle \int \frac{dx}{\sqrt{\ln(x)}}$ goes to $0$ slower than $\displaystyle \int \frac{dx}{\sqrt{x}}$, thus from integral comparison: $$\displaystyle \int_{1}^{2}\frac{dx}{\sqrt{(2-x)\ln(x)}} < \int_{1}^{2} \frac{dx}{\sqrt{x}}$$ it diverges. Is this correct? Thanks.