Proving $\;\ln k \geq \int_{k-\frac{1}{2}}^{k+ \frac{1}{2}}\ln x dx$

I'm trying to prove $$\ln k \geq \int_{k-\frac{1}{2}}^{k+ \frac{1}{2}}\ln x dx$$

In other words, I'm trying to show why the area of the rectangle with height $\ln k$ and width $1$ bounds the area under the graph of $f(x)=\ln x$ in the interval $[k-\frac{1}{2},k+\frac{1}{2}].$

I tried to integrate but got stuck. Any ideas for an elegant proof for this?

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