Answer
Divergent
Work Step by Step
$$\int_{1}^{\infty }\frac{lnx}{x}dx=\lim_{t\rightarrow \infty}\int_{1}^{t}\frac{lnx}{x}dx$$
$$=\lim_{t\rightarrow \infty}\int_{1}^{t}lnx\ d(lnx)$$
$$=\lim_{t\rightarrow \infty} \left|\frac{(lnx)^{2}}{2}\right |_{1}^{t}$$
$$= \lim_{t\rightarrow \infty} \frac{(ln\,t)^{2}}{2}$$
$$=\infty,\ divergent$$
