I have to prove that:

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> If $r > 1$ then $\log r - \int\_{r}^{r+1}\log(x)\, dx$ differs from $-\frac{1}{2r}$ by less than $\frac{1}{6r^2}$.
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I am thinking about working out the integral using Taylor Series for log(1+x) at the point 0. This seems like it will work, but when I attempt it I do not get the correct statement. Any hints or alternate proofs that would work?