Let's solve the integral step by step: \[ \int \frac{x}{x-2} dx \] ### Step 1: **Substitution** Let \( u = x - 2 \) (as given in the hint), so \( x = u + 2 \) and \( dx = du \). ### Step 2: **Rewrite the Integral** Substitute \( x \) and \( dx \): \[ \int \frac{x}{x-2} dx = \int \frac{u + 2}{u} du \] ### Step 3: **Simplify the Expression** \[ \frac{u + 2}{u} = 1 + \frac{2}{u} \] So the integral becomes: \[ \int \left(1 + \frac{2}{u}\right) du \] ### Step 4: **Integrate Term by Term** \[ \int 1 \, du + \int \frac{2}{u} du = u + 2 \ln|u| + C \] ### Step 5: **Substitute Back for \( u \)** Recall that \( u = x - 2 \): \[ u + 2 \ln|u| + C = (x - 2) + 2\ln|x-2| + C \] --- ### **Final Answer** \[ \boxed{(x - 2) + 2\ln|x-2| + C} \]

To evaluate the integral \(\int \frac{x}{x-2} dx\), we use substitution \(u = x - 2\), simplifying the integrand and then integrating term by term. This method transforms the integral into a basic form involving a linear and a logarithmic term.