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Math Help - Riemann -Stieltjes integral

  1. #1
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    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).
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  2. #2
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    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?
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