1. **State the problem:** We need to evaluate the integral $$\int \frac{x^2}{x-1} \, dx$$. 2. **Formula and approach:** When integrating a rational function where the degree of the numerator is greater than or equal to the degree of the denominator, we first perform polynomial division. 3. **Perform polynomial division:** Divide $x^2$ by $x-1$. $$x^2 \div (x-1) = x + 1 + \frac{1}{x-1}$$ Explanation: $x^2 = (x-1)(x+1) + 1$. 4. **Rewrite the integral:** $$\int \frac{x^2}{x-1} \, dx = \int \left(x + 1 + \frac{1}{x-1}\right) dx$$ 5. **Integrate term-by-term:** $$\int x \, dx + \int 1 \, dx + \int \frac{1}{x-1} \, dx = \frac{x^2}{2} + x + \ln|x-1| + C$$ 6. **Final answer:** $$\int \frac{x^2}{x-1} \, dx = \frac{x^2}{2} + x + \ln|x-1| + C$$ This completes the integration by polynomial division and term-wise integration.