Question

Compute \(\int \log x \, dx\)

• A. \(\log x - x + C\)
• B. \(x \log x - x + C\)
• C. \(\log x + x + C\)
• D. \(x \log x + x + C\)

Hint:

To verify any integration result quickly, differentiate the options.
Taking the derivative of Option (B):
\[ \frac{d}{dx}(x \log x - x + C) = \left( 1 \cdot \log x + x \cdot \frac{1}{x} \right) - 1 = \log x + 1 - 1 = \log x \] This confirms the result immediately.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Integrating the natural logarithm function, \(\log x\) (or \(\ln x\)), requires the integration by parts method.
This technique is based on the product rule of differentiation and helps integrate the product of two functions.
Key Formula or Approach:
The formula for integration by parts is:
\[ \int u \, dv = u v - \int v \, du \] According to the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential), we select the logarithmic function as \(u\) and the constant function \(1\) as the remaining part \(dv\).

Step 2: Detailed Explanation:
Let us perform the integration step-by-step:
1. Let:
\[ u = \log x \] \[ dv = dx \] 2. Differentiate \(u\) to find \(du\):
\[ du = \frac{1}{x} \, dx \] 3. Integrate \(dv\) to find \(v\):
\[ v = x \] 4. Substitute \(u\), \(v\), \(du\), and \(dv\) into the integration by parts formula:
\[ \int \log x \, dx = (\log x)(x) - \int (x) \left( \frac{1}{x} \, dx \right) \] 5. Simplify the expression inside the integral:
\[ \int \log x \, dx = x \log x - \int 1 \, dx \] 6. Complete the integration:
\[ \int \log x \, dx = x \log x - x + C \] Where \(C\) is the constant of integration.

Step 3: Final Answer:
The correct option is (B).