How do I show that $\int\_1^\infty\frac 1{\ln x}dx$ diverges? I'm thinking to break up the integral into two parts, say, over $[1,2]$ and $[2,\infty)$, but how do I integrate the integrand?

I'm stumped partly because the author claims part (f) is proved similarly as part (a)/(b). But how can this be!? Part (f) is so much more non-trivial than (a)/(b).

![enter image description here](https://i.stack.imgur.com/UouPg.jpg)