<p>Evaluate \(\displaystyle\int_1^e \frac{\ln x}{x}\,dx\)</p>
Step-by-Step Solution
Key Concept: Let t = ln x, dt = dx/x. Limits: x=1 \to t=0; x=e \to t=1.
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<p>Let \(t = \ln x \Rightarrow dt = \dfrac{dx}{x}\).</p>
<p>When \(x=1: t=0\); when \(x=e: t=1\).</p>
<p>\[\int_1^e\frac{\ln x}{x}\,dx = \int_0^1 t\,dt = \left[\frac{t^2}{2}\right]_0^1 = \boxed{\frac{1}{2}}\]</p>
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Correct Answer: A