Question

The integration of $\log x$ with respect to $x$ is

• A. $\frac{1}{x} + C$
• B. $x\log x - x + C$
• C. $x\log x + x + C$
• D. $\frac{\log x}{x} + C$

Hint:

Integration by parts is useful when the integrand is a product of algebraic and logarithmic functions.

Answer 1

The Correct Option is B

Step 1: Use integration by parts.
Let $u = \log x$, $dv = dx$.
Step 2: Differentiate and integrate.
\[ du = \frac{1}{x}dx,\quad v = x \]
Step 3: Apply the formula.
\[ \int u\,dv = uv - \int v\,du \]
Step 4: Compute the integral.
\[ \int \log x\,dx = x\log x - \int x\cdot\frac{1}{x}dx \] \[ = x\log x - \int 1\,dx \] \[ = x\log x - x + C \]