1. **State the problem:** Find the integral $$\int (x^3 + 1) \, dx$$. 2. **Recall the integral rule:** The integral of $$x^n$$ with respect to $$x$$ is $$\frac{x^{n+1}}{n+1} + C$$, where $$n \neq -1$$. 3. **Apply the rule to each term:** $$\int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4}$$ $$\int 1 \, dx = x$$ 4. **Combine the results:** $$\int (x^3 + 1) \, dx = \frac{x^4}{4} + x + C$$ 5. **Explain the constant:** The $$+ C$$ represents the constant of integration, accounting for any constant term lost during differentiation. **Final answer:** $$\boxed{\frac{x^4}{4} + x + C}$$