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Question
Solution
\[Let\ I = \int_1^e \frac{\log x}{x} d x\]
\[Let\ \log x = u\]
\[ \Rightarrow \frac{1}{x} dx = du\]
\[ \therefore I = \int u\ d u\]
\[ \Rightarrow I = \left[ \frac{u^2}{2} \right]\]
\[ \Rightarrow I = \left[ \frac{(\log x )^2}{2} \right]_1^e \]
\[ \Rightarrow I = \frac{1}{2} - 0\]
\[ \Rightarrow I = \frac{1}{2}\]
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