Step 1: Understanding the Concept:
This is a definite integral that can be solved using the method of substitution. The presence of both \( \log x \) and its derivative \( 1/x \) makes substitution straightforward.
: Key Formula or Approach:
Use the substitution \( u = \log x \). Then, \( du = \frac{1}{x} dx \).
Step 2: Detailed Explanation:
Let \( I = \int_{1}^{e} \frac{\log x}{x} dx \).
Substitute \( u = \log x \).
When \( x = 1 \), \( u = \log 1 = 0 \).
When \( x = e \), \( u = \log e = 1 \).
The integral becomes:
\[ I = \int_{0}^{1} u \, du \]
\[ I = \left[ \frac{u^{2}}{2} \right]_{0}^{1} \]
\[ I = \frac{1^{2}}{2} - \frac{0^{2}}{2} = \frac{1}{2} \].
Step 3: Final Answer:
The value of the integral is 1/2.