Question Detail

To find $\int \ln \frac{x}{2} dx$, we use integration by parts, $\int u \, dv = uv - \int v \, du$.Let $u = \ln \frac{x}{2}$ and $dv = dx$.Then $du = \frac{1}{x/2} \cdot \frac{1}{2} dx = \frac{2}{x} \cdot \frac{1}{2} dx = \frac{1}{x} dx$, and $v = x$.So, $\int \ln \frac{x}{2} dx = x \ln \frac{x}{2} - \int x \cdot \frac{1}{x} dx$$= x \ln \frac{x}{2} - \int 1 \, dx$$= x \ln \frac{x}{2} - x + C$$= x (\ln \frac{x}{2} - 1) + C$.Thus, option D is the correct answer.