1. The problem is to find the integral of the function $2x - 2.7x + 7$ with respect to $x$ and verify if it equals $\frac{1}{4}x^4 - 7x + 1$. 2. First, simplify the integrand: $$2x - 2.7x + 7 = (2 - 2.7)x + 7 = -0.7x + 7$$ 3. The integral of a function $f(x)$ is given by: $$\int f(x)\,dx = F(x) + C$$ where $F'(x) = f(x)$ and $C$ is the constant of integration. 4. Integrate each term separately: $$\int (-0.7x)\,dx = -0.7 \int x\,dx = -0.7 \cdot \frac{x^2}{2} = -0.35x^2$$ $$\int 7\,dx = 7x$$ 5. Combine the results: $$f(x) = -0.35x^2 + 7x + C$$ 6. Compare this with the given function $\frac{1}{4}x^4 - 7x + 1$: - The integral we found is a quadratic function, but the given function is quartic ($x^4$ term). - Therefore, the given function is not the integral of $2x - 2.7x + 7$. Final answer: $$\int (2x - 2.7x + 7)\,dx = -0.35x^2 + 7x + C$$