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Answer: \(3\log 3 - 2\)

Integrate by parts, taking \(u = \log x\) and \(v = 1\):

\(\displaystyle\int \log x\,dx = \log x \int 1\,dx - \int\left[\frac{d}{dx}(\log x)\int 1\,dx\right]dx = x\log x - \int \frac{1}{x} \cdot x\,dx = x\log x - x\)

\(\displaystyle\int_1^3 \log x\,dx = \Big[x\log x - x\Big]_1^3 = (3\log 3 - 3) - (1 \cdot \log 1 - 1) = 3\log 3 - 3 - 0 + 1 = 3\log 3 - 2\)

Doubtly AI · 29 Oct 2026 AI-verified · Nov 2026 Report
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