I was trying to get an asymptotic formula for the sum
$$\sum\_{n=2}^{x}\frac{1}{\log n}$$
this looks a bit tricky, but I still tried it. We have

\begin{align}
\sum\_{n=2}^{x}\frac{1}{\log n}&=-\sum\_{n=2}^{x}\frac{1}{\log(nt)}\Bigg{|}\_{t=1}^{t=\infty}\\
&=\sum\_{n=2}^{x}\int\_{1}^{\infty}\frac1{t\log^2(nt)}dt
\end{align}
Now I don't know what to do further. Any help would be appreciated. Any asymptotic formula or closed form would help.  

**Note:** I will probably update this question to post more of my work.  

**Update:** I proved that this sum is $O(x/\log x)$, and this was also mentioned in the comments. But I want to improve this. How can one improve the error term?