1. We are asked to find the integral of the function $x^2 + 5x$ with respect to $x$. 2. The formula for integrating a sum is the sum of the integrals: $$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$$ 3. We use the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$. 4. Applying the power rule to each term: - For $x^2$: $$\int x^2 \, dx = \frac{x^{3}}{3}$$ - For $5x$: $$\int 5x \, dx = 5 \int x \, dx = 5 \cdot \frac{x^{2}}{2} = \frac{5x^{2}}{2}$$ 5. Adding the results together: $$\int (x^2 + 5x) \, dx = \frac{x^{3}}{3} + \frac{5x^{2}}{2} + C$$ 6. Here, $C$ is the constant of integration representing any constant value. Final answer: $$\boxed{\frac{x^{3}}{3} + \frac{5x^{2}}{2} + C}$$