Question Detail

To integrate (\frac{x}{x-1}), we can use algebraic manipulation or substitution.Method 1: Algebraic manipulation(\frac{x}{x-1} = \frac{(x-1)+1}{x-1} = 1 + \frac{1}{x-1})So, (\int \frac{x}{x-1}\,dx = \int \left(1 + \frac{1}{x-1}\right)\,dx)(= \int 1\,dx + \int \frac{1}{x-1}\,dx)(= x + \ln|x-1| + C).The option given is (x + \ln(x-1)), implying (x-1 > 0) or considering the general form.