Question

Evaluate: \[ \int \log x \,dx. \]

Hint:

Use integration by parts where \( u = \log x \) and \( dv = dx \) for \(\int \log x \,dx\).

Answer 1

Step 1: Use Integration by Parts
Using the formula: \[ \int u \, dv = u v - \int v \, du, \] let: \[ u = \log x, \quad dv = dx. \] Step 2: Compute \( du \) and \( v \)
\[ du = \frac{1}{x} dx, \quad v = x. \] Step 3: Apply Integration by Parts
\[ \int \log x \,dx = x \log x - \int x \times \frac{1}{x} dx. \] \[ = x \log x - \int dx. \] \[ = x \log x - x + C. \]