1. The problem is to evaluate the definite integral $$\int_1^7 x \, dx$$. 2. The formula for the integral of $$x$$ is $$\int x \, dx = \frac{x^2}{2} + C$$, where $$C$$ is the constant of integration. 3. 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. Applying this to our problem: $$F(x) = \frac{x^2}{2}$$ 5. Evaluate at the limits: $$F(7) = \frac{7^2}{2} = \frac{49}{2}$$ $$F(1) = \frac{1^2}{2} = \frac{1}{2}$$ 6. Subtract to find the definite integral: $$\int_1^7 x \, dx = F(7) - F(1) = \frac{49}{2} - \frac{1}{2} = \frac{48}{2}$$ 7. Simplify the fraction: $$\frac{48}{2} = 24$$ 8. Therefore, the value of the integral is $$24$$.