1. **Stating the problem:** We want to find the integral $$I_n = \int (1 - x)^n \, dx$$ where $n$ is a constant exponent. 2. **Formula and rules:** To integrate powers of a linear function, we use the substitution method or the power rule for integrals. The power rule states: $$\int u^n \, du = \frac{u^{n+1}}{n+1} + C \quad \text{for } n \neq -1$$ 3. **Substitution:** Let $$u = 1 - x$$ then $$du = -dx$$ or $$dx = -du$$. 4. **Rewrite the integral:** $$I_n = \int (1 - x)^n \, dx = \int u^n (-du) = - \int u^n \, du$$ 5. **Integrate:** $$- \int u^n \, du = - \frac{u^{n+1}}{n+1} + C = - \frac{(1 - x)^{n+1}}{n+1} + C$$ 6. **Final answer:** $$I_n = - \frac{(1 - x)^{n+1}}{n+1} + C$$ This formula holds for all $n \neq -1$. If $n = -1$, the integral becomes $$\int \frac{1}{1-x} dx = -\ln|1-x| + C$$.