Log integral inequality

Question

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Attempt:

$\int_{1}^{n}ln(x)dx = \int_{1}^{3/2}ln(x)dx+\int_{3/2}^{5/2}ln(x)dx+\int_{5/2}^{7/2}ln(x)dx+...+\int_{n-\frac{1}{2}}^{n}ln(x)dx$

$≤ \frac{1}{8}+ln(2)+ln(3)+...$