Question

determine whether improper integral converges, if so, evaluate it.

Answer 1

\[\int\limits_{1}^{2}dx/xlnx\]

Answer 2

\[\int\limits_{1}^{2} \frac{dx}{xln(x)}\rightarrow u=lnx, du=\frac{1}{x}dx\]
Change the limits: upper=ln2, lower=0
\[\lim_{t \rightarrow 0^+} \int\limits_{t}^{\ln2} \frac{du}{u} \rightarrow \lim_{t \rightarrow 0^+} \left[ lnu \right]_{t}^{\ln2}\rightarrow \lim_{t \rightarrow 0^+} \left[ \ln(\ln2)-\ln(t) \right] =ln(ln2) -\infty\]
so it diverges