1. **State the problem:** Evaluate the expression $$1 + \int (1 - x^3) - (1 - x) \, dx$$.\n\n2. **Rewrite the integral:** The integral expression is ambiguous without limits, so we assume it is an indefinite integral. The integrand is $$(1 - x^3) - (1 - x)$$. Simplify the integrand:\n$$ (1 - x^3) - (1 - x) = 1 - x^3 - 1 + x = x - x^3 $$\n\n3. **Integral formula:** The integral of a sum/difference is the sum/difference of the integrals. Use the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$.\n\n4. **Integrate term-by-term:**\n$$\int (x - x^3) \, dx = \int x \, dx - \int x^3 \, dx = \frac{x^2}{2} - \frac{x^4}{4} + C$$\n\n5. **Add the constant term outside the integral:**\n$$1 + \left( \frac{x^2}{2} - \frac{x^4}{4} + C \right) = 1 + \frac{x^2}{2} - \frac{x^4}{4} + C$$\n\n6. **Final answer:** The indefinite integral plus 1 is\n$$\boxed{1 + \frac{x^2}{2} - \frac{x^4}{4} + C}$$\nwhere $C$ is the constant of integration.