Question Detail

To integrate $\frac{x}{x-1}$, we can use algebraic manipulation or polynomial long division.Rewrite the numerator as $(x-1)+1$:$\frac{x}{x-1} = \frac{(x-1)+1}{x-1} = \frac{x-1}{x-1} + \frac{1}{x-1} = 1 + \frac{1}{x-1}$.Now integrate term by term:$\int \left(1 + \frac{1}{x-1}\right) dx = \int 1 \, dx + \int \frac{1}{x-1} \, dx$$= x + \ln|x-1| + C$.Thus, option B is the correct answer.