Concept:
Integrating functions that do not have a straightforward antiderivative often requires the Integration by Parts method.
Step 1: Understanding the Question:
We want to find the indefinite integral of the natural logarithm of \(x\). Since \(\log x\) is not the derivative of a basic function, we treat it as a product: \(\log x \cdot 1\).
Step 2: Key Formula or Approach:
Integration by Parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential), we choose:
- \(u = \log x\)
- \(dv = dx\)
Step 3: Detailed Solution:
1. Let \(u = \log x \implies du = \frac{1}{x} dx\).
2. Let \(dv = dx \implies v = x\).
3. Apply the formula:
\[ \int \log x \, dx = (\log x)(x) - \int x \left(\frac{1}{x} dx\right) \]
\[ = x \log x - \int 1 \, dx \]
\[ = x \log x - x + C \]
Step 4: Final Answer:
The integral is \(x \log x - x + C\).