Question

Integrate \( \log_e x \) with respect to \( x \).

Hint:

When integrating \( \log_e x \), use integration by parts with \( u = \log_e x \) and \( dv = dx \).

Answer 1

Step 1: Use integration by parts. Let: - \( u = \log_e x \), so \( du = \frac{1}{x} \, dx \) - \( dv = dx \), so \( v = x \) 

Step 2: Apply the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] Substitute the values: \[ \int \log_e x \, dx = x \log_e x - \int x \cdot \frac{1}{x} \, dx \] 

Step 3: Simplify: \[ = x \log_e x - \int 1 \, dx = x \log_e x - x \]