1. **State the problem:** Calculate the definite integral $$\int_2^8 x \, dx$$. 2. **Formula and rules:** The integral of $$x$$ with respect to $$x$$ is given by $$\int x \, dx = \frac{x^2}{2} + C$$, where $$C$$ is the constant of integration. 3. **Apply the definite integral formula:** For definite integrals, we evaluate the antiderivative at the upper and lower limits and subtract: $$\int_a^b f(x) \, dx = F(b) - F(a)$$ where $$F(x)$$ is the antiderivative of $$f(x)$$. 4. **Calculate the antiderivative:** $$F(x) = \frac{x^2}{2}$$ 5. **Evaluate at the limits:** $$F(8) = \frac{8^2}{2} = \frac{64}{2} = 32$$ $$F(2) = \frac{2^2}{2} = \frac{4}{2} = 2$$ 6. **Subtract to find the definite integral:** $$\int_2^8 x \, dx = F(8) - F(2) = 32 - 2 = 30$$ **Final answer:** $$30$$