1. **State the problem:** Evaluate the integral $$\int (1 - x) \sqrt{x} \, dx.$$\n\n2. **Rewrite the integrand:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}.$$ So the integral becomes $$\int (1 - x) x^{\frac{1}{2}} \, dx = \int \left(x^{\frac{1}{2}} - x^{\frac{3}{2}}\right) \, dx.$$\n\n3. **Use the power rule for integration:** For any real number $$n \neq -1$$, $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$$\n\n4. **Integrate each term separately:**\n$$\int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} x^{\frac{3}{2}}.$$\n$$\int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}}.$$\n\n5. **Combine the results:**\n$$\int (1 - x) \sqrt{x} \, dx = \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C.$$\n\n6. **Final answer:**\n$$\boxed{\frac{2}{3} x^{\frac{3}{2}} - \frac{2}{5} x^{\frac{5}{2}} + C}.$$