1. Stating the problem: We need to evaluate the integral $$\int \left(\sqrt{x} + \sqrt{a} \sqrt{x}\right) \, dx$$ where $a$ is a constant. 2. Rewrite the integrand using exponent notation: Recall that $\sqrt{x} = x^{\frac{1}{2}}$ and $\sqrt{a} = a^{\frac{1}{2}}$. So the integrand becomes $$x^{\frac{1}{2}} + a^{\frac{1}{2}} x^{\frac{1}{2}} = \left(1 + a^{\frac{1}{2}}\right) x^{\frac{1}{2}}$$ 3. Use the linearity of the integral: $$\int \left(1 + a^{\frac{1}{2}}\right) x^{\frac{1}{2}} \, dx = \left(1 + a^{\frac{1}{2}}\right) \int x^{\frac{1}{2}} \, dx$$ 4. Apply the power rule for integration: For $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n \neq -1$. 5. Calculate the integral: $$\int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} x^{\frac{3}{2}} + C$$ 6. Substitute back: $$\left(1 + a^{\frac{1}{2}}\right) \times \frac{2}{3} x^{\frac{3}{2}} + C = \frac{2}{3} \left(1 + \sqrt{a}\right) x^{\frac{3}{2}} + C$$ Final answer: $$\int \left(\sqrt{x} + \sqrt{a} \sqrt{x}\right) \, dx = \frac{2}{3} \left(1 + \sqrt{a}\right) x^{\frac{3}{2}} + C$$