Question:
\(\int_1^e \frac{\log x}{x} \, dx\) is
\(\int_1^e \frac{\log x}{x} \, dx\) is
Show Hint
Whenever you see \(\frac{\log x}{x}\), directly think derivative of \((\log x)^2\).
Updated On: Apr 15, 2026
- \( \frac{1}{2} \)
- \(1\)
- \(e\)
- None of these
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The Correct Option is A
Solution and Explanation
Concept:
\[
\int \frac{\log x}{x}\,dx = \frac{(\log x)^2}{2}
\]
Step 1: Apply limits. \[ \int_1^e \frac{\log x}{x} \, dx = \left[\frac{(\log x)^2}{2}\right]_1^e \]
Step 2: Evaluate. \[ = \frac{(\log e)^2}{2} - \frac{(\log 1)^2}{2} = \frac{1^2}{2} - 0 = \frac{1}{2} \]
Step 1: Apply limits. \[ \int_1^e \frac{\log x}{x} \, dx = \left[\frac{(\log x)^2}{2}\right]_1^e \]
Step 2: Evaluate. \[ = \frac{(\log e)^2}{2} - \frac{(\log 1)^2}{2} = \frac{1^2}{2} - 0 = \frac{1}{2} \]
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