Does $\int_{1}^{2}\frac{dx}{\sqrt{(2-x)\ln(x)}}$ converge? 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.

I solve the improper integral firstly by take it apart at $3/2$:
$$\int_1^{3/2}f(x)dx+\int_{3/2}^2f(x)dx$$
Now in the first integral we see that $$\lim_{x\to 1^+}\sqrt{x-1}f(x)=1<\infty$$ And for the second one we get $$\lim_{x\to 2^-}\sqrt{2-x}f(x)=1/\sqrt{\ln 2}<\infty$$ Therefore, both are converge and so the summation converges. 
