So I am trying to show the following: $$\sum_{n \leq x} \frac{\mu(n)}{n} \log{\frac{x}{n}}=O(1) $$ so I tried partial summation as following:
Let $A(x)=\sum_{n \leq x} \frac{\mu(n)}{n}$, then we have $$\sum_{n \leq x} \frac{\mu(n)}{n} \log{\frac{x}{n}}= \int_1^{x} \frac{A(t)\mathrm dt}{t},$$ and $A(t)$ is clearly very small, or $o(1)$ for all $t \in [1,x]$. My question is how to go from here to conclude that the error term is $O(1)$?