1. Stated problem: Calculate the integral $$\int (x^2 + 5x) \, dx$$. 2. Formula and rules: The integral of a sum is the sum of the integrals, and the power rule for integration states $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for any real number $n \neq -1$. 3. Intermediate work: - Split the integral: $$\int x^2 \, dx + \int 5x \, dx$$ - Integrate each term: $$\int x^2 \, dx = \frac{x^{3}}{3} + C_1$$ $$\int 5x \, dx = 5 \int x \, dx = 5 \cdot \frac{x^{2}}{2} + C_2 = \frac{5x^{2}}{2} + C_2$$ 4. Combine results: $$\int (x^2 + 5x) \, dx = \frac{x^{3}}{3} + \frac{5x^{2}}{2} + C$$ 5. Explanation: We used the linearity of integrals to separate the terms and applied the power rule to each term. Constants multiply the integral directly. Final answer: $$\boxed{\frac{x^{3}}{3} + \frac{5x^{2}}{2} + C}$$