Math Help - Riemann -Stieltjes integral

1. Riemann -Stieltjes integral

Let $h(x)=x+[x]-\frac{1}{2}$ if $x$ is not in $Z$ and $h(x)=0$ if $x \in Z$.
Find the value of $\int_1 ^n log(x)dh(x)$.

2. Integral

This is what i got so far. i am not sure if this is correct at all since i am not sure how $h(x)=0$ if $x \in Z$ affects.

$\int log(x)dh(x)= \int log(x)d(x-[x]-\frac{1}{2})= \int log(x)dx - \int log(x)d([x])-\int log(x) d(\frac{1}{2})$.

$\int_1^n log(x)dx=xlog(x)-x \mid_1^n = nlog(n)-n+1$
$\int log(x)d([x])= \sum_{x=1}^{[n] }log(x)$ since the jump at each interger is 1.
$\int log(x)d(\frac{1}{2})=0$
so i got $\int log(x)dh(x)= nlog(n)-n+1-\sum_{x=1}^{[n]} log(x)$.

Can anyone comfirm or have any suggestioin?