1. **State the problem:** We want to evaluate the integral $$\int \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x}} \, dx$$ where $a$ is a constant. 2. **Simplify the integrand:** Use the property of square roots and division: $$\frac{\sqrt{x} + \sqrt{a}}{\sqrt{x}} = \frac{\sqrt{x}}{\sqrt{x}} + \frac{\sqrt{a}}{\sqrt{x}} = 1 + \sqrt{a} \cdot \frac{1}{\sqrt{x}}$$ 3. **Rewrite the integrand:** $$1 + \sqrt{a} x^{-\frac{1}{2}}$$ 4. **Set up the integral:** $$\int \left(1 + \sqrt{a} x^{-\frac{1}{2}}\right) dx = \int 1 \, dx + \sqrt{a} \int x^{-\frac{1}{2}} \, dx$$ 5. **Integrate each term:** - Integral of 1 with respect to $x$ is $x$. - Integral of $x^{-\frac{1}{2}}$ is: $$\int x^{-\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} = 2\sqrt{x}$$ 6. **Combine results:** $$x + \sqrt{a} \cdot 2\sqrt{x} + C = x + 2\sqrt{a}\sqrt{x} + C$$ 7. **Final answer:** $$\boxed{x + 2\sqrt{a}\sqrt{x} + C}$$ This is the evaluated integral of the given expression.