1. Problem: Calculate the indefinite integral $$\int (x + x^4 + 2x^5) \, dx$$. 2. Formula: The integral of a sum is the sum of the integrals, and the integral of $$x^n$$ is $$\frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. 3. Apply the formula to each term: - $$\int x \, dx = \frac{x^{1+1}}{1+1} = \frac{x^2}{2}$$ - $$\int x^4 \, dx = \frac{x^{4+1}}{4+1} = \frac{x^5}{5}$$ - $$\int 2x^5 \, dx = 2 \int x^5 \, dx = 2 \cdot \frac{x^{5+1}}{5+1} = 2 \cdot \frac{x^6}{6} = \frac{2x^6}{6} = \frac{x^6}{3}$$ 4. Combine the results: $$\int (x + x^4 + 2x^5) \, dx = \frac{x^2}{2} + \frac{x^5}{5} + \frac{x^6}{3} + C$$ 5. Final answer: $$\boxed{\frac{x^2}{2} + \frac{x^5}{5} + \frac{x^6}{3} + C}$$