1. **State the problem:** We need to evaluate the integral $$\int (\sqrt{x} + 1)(x - \sqrt{x} + 1) \, dx$$. 2. **Expand the integrand:** Multiply the two binomials: $$ (\sqrt{x} + 1)(x - \sqrt{x} + 1) = \sqrt{x} \cdot x + \sqrt{x} \cdot (-\sqrt{x}) + \sqrt{x} \cdot 1 + 1 \cdot x + 1 \cdot (-\sqrt{x}) + 1 \cdot 1 $$ 3. **Simplify each term:** - $\sqrt{x} \cdot x = x^{3/2}$ - $\sqrt{x} \cdot (-\sqrt{x}) = -x$ - $\sqrt{x} \cdot 1 = \sqrt{x}$ - $1 \cdot x = x$ - $1 \cdot (-\sqrt{x}) = -\sqrt{x}$ - $1 \cdot 1 = 1$ 4. **Combine like terms:** $$ x^{3/2} - x + \sqrt{x} + x - \sqrt{x} + 1 = x^{3/2} + 1 $$ 5. **Rewrite the integral:** $$ \int (x^{3/2} + 1) \, dx $$ 6. **Integrate term-by-term:** - Integral of $x^{3/2}$ is $$ \frac{x^{5/2}}{\frac{5}{2}} = \frac{2}{5} x^{5/2} $$ - Integral of $1$ is $x$ 7. **Write the final answer:** $$ \int (\sqrt{x} + 1)(x - \sqrt{x} + 1) \, dx = \frac{2}{5} x^{5/2} + x + C $$ where $C$ is the constant of integration.