1. The problem is to evaluate the definite integral $$\int_2^7 x \, dx$$. 2. The formula for the definite integral of a function $f(x)$ from $a$ to $b$ is: $$\int_a^b f(x) \, dx = F(b) - F(a)$$ where $F(x)$ is the antiderivative of $f(x)$. 3. For $f(x) = x$, the antiderivative is: $$F(x) = \frac{x^2}{2}$$ 4. Evaluate $F(x)$ at the bounds: $$F(7) = \frac{7^2}{2} = \frac{49}{2}$$ $$F(2) = \frac{2^2}{2} = \frac{4}{2} = 2$$ 5. Calculate the definite integral: $$\int_2^7 x \, dx = F(7) - F(2) = \frac{49}{2} - 2 = \frac{49}{2} - \frac{4}{2} = \frac{45}{2} = 22.5$$ The value of the integral $$\int_2^7 x \, dx$$ is **22.5**.