Question

Integration of \(\ln(x)\) with \(x\), i.e. \(\int \ln(x)dx =\) __________.
 

• A. \(x \cdot \ln(x) - x + {Constant}\)
• B. \(x - \ln(x) + {Constant}\)
• C. \(x \cdot \ln(x) + x + {Constant}\)
• D. \(\ln(x) - x + {Constant}\)

Hint:

When integrating logarithmic functions like \( \ln(x) \), consider integration by parts with \( u = \ln(x) \). This technique often simplifies otherwise difficult integrals.

Answer 1

The Correct Option is A

To integrate \( \int \ln(x) dx \), we use the method of integration by parts. Recall the formula: \[ \int u\,dv = uv - \int v\,du \] Choose: \[ u = \ln(x) \Rightarrow du = \frac{1}{x} dx,\quad dv = dx \Rightarrow v = x \] Now apply the integration by parts formula: \[ \int \ln(x)\, dx = x \cdot \ln(x) - \int x \cdot \frac{1}{x}\, dx \] \[ = x \cdot \ln(x) - \int 1\, dx = x \cdot \ln(x) - x + C \] Therefore, \[ \int \ln(x)\, dx = x \cdot \ln(x) - x + {Constant} \] This matches option (A).