1. **State the problem:** Evaluate the integral $$\int \frac{(1-x)^2}{x \sqrt{x}} \, dx$$. 2. **Rewrite the integrand:** Note that $$\sqrt{x} = x^{1/2}$$, so the denominator is $$x \cdot x^{1/2} = x^{3/2}$$. 3. **Simplify the integrand:** $$\frac{(1-x)^2}{x \sqrt{x}} = \frac{(1-x)^2}{x^{3/2}} = (1-x)^2 x^{-3/2}$$. 4. **Expand the numerator:** $$(1-x)^2 = 1 - 2x + x^2$$. 5. **Rewrite the integrand as a sum:** $$ (1 - 2x + x^2) x^{-3/2} = x^{-3/2} - 2x^{-1/2} + x^{1/2}$$. 6. **Integrate term-by-term:** - $$\int x^{-3/2} dx = \int x^{-\frac{3}{2}} dx = \frac{x^{-\frac{3}{2}+1}}{-\frac{3}{2}+1} + C = \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C = -2x^{-1/2} + C$$. - $$\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} + C = 2x^{1/2} + C$$. - $$\int x^{1/2} dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C$$. 7. **Combine the results:** $$\int \frac{(1-x)^2}{x \sqrt{x}} dx = -2x^{-1/2} - 2(2x^{1/2}) + \frac{2}{3} x^{3/2} + C = -2x^{-1/2} - 4x^{1/2} + \frac{2}{3} x^{3/2} + C$$. 8. **Final answer:** $$\boxed{-2x^{-1/2} - 4x^{1/2} + \frac{2}{3} x^{3/2} + C}$$