Question

Evaluate \( \int \log x \, dx \).

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

Hint:

Whenever you see \( \int \log x \, dx \), immediately think of integration by parts. Choose \(u = \log x\) because its derivative becomes simpler \((1/x)\).

Answer 1

The Correct Option is A

Concept: To evaluate the integral of \( \log x \), we use the method of integration by parts. The formula for integration by parts is: \[ \int u\,dv = uv - \int v\,du \] This method is useful when the integrand is a product of two functions.

Step 1: Choose appropriate functions. Let \[ u = \log x, \qquad dv = dx \] Then \[ du = \frac{1}{x}dx, \qquad v = x \]

Step 2: Apply the integration by parts formula. \[ \int \log x\,dx = x\log x - \int x \cdot \frac{1}{x} dx \] \[ = x\log x - \int 1\,dx \]

Step 3: Complete the integration. \[ \int 1\,dx = x \] Therefore, \[ \int \log x\,dx = x\log x - x + C \] \[ \boxed{x\log x - x + C} \]