1. Stating the problem: Evaluate the definite integral $$\int_0^2 \frac{x^2}{x} \, dx$$. 2. Simplify the integrand: Since $$\frac{x^2}{x} = x$$ for $$x \neq 0$$, the integral becomes $$\int_0^2 x \, dx$$. 3. Use the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. 4. Apply the rule with $$n=1$$: $$\int x \, dx = \frac{x^{2}}{2} + C$$. 5. Evaluate the definite integral from 0 to 2: $$\left[ \frac{x^{2}}{2} \right]_0^2 = \frac{2^{2}}{2} - \frac{0^{2}}{2} = \frac{4}{2} - 0 = 2$$. 6. Final answer: $$2$$.