The integral:
$$\int_2^\infty \frac1{\ln(x)^p}\,dx $$
I have no idea on how to find a solution to this. It just seems impossible to have a solution to this integral in which it converges, however, my teacher insists that it does, even though he refuses to do it himself. Is that true? Then how can I do it?
Hint: with asymptotic analysis:
As for any $p$, we have $\ln^p x=_\infty o(x)$, we also have $\dfrac1x=_{\infty} o\Bigl(\dfrac 1{\ln^p x}\Bigr)$, so if $x$ is large enough, $$\frac 1{\ln^p x}>\frac 1x.$$