Another exercise from Apostol's book, this time we're supposed to prove
$\mathrm{Li}(x)=\frac{x}{\log x}+\int_2^x \frac{dt}{\log^2t}-\frac{2}{\log 2}.$
which is easy to do via integration by parts. But then he goes on to write,
[...] and that, more generally, $\mathrm{Li}(x)=\frac{x}{\log x} \left(1+ \sum_{k=1}^{n-1} \frac{k!}{\log^k x} \right)+n! \int_2^x \frac{dt}{\log^{n+1}t}+C_n,$ where $C_n$ is independent of $x$.
My question may come across as stupid, but what's going on here? Is this a comment, or should I be able to prove this as well? If so, I'd very much like some small hint on how to approach this! Thanks!