1. **State the problem:** We need to evaluate the integral $$\int \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x}} \, dx$$ where $a$ is a constant. 2. **Simplify the integrand:** Split the fraction into two parts: $$\frac{\sqrt{x}}{\sqrt{x}} + \frac{\sqrt{a}}{\sqrt{x}} = 1 + \sqrt{a} \cdot x^{-\frac{1}{2}}$$ 3. **Rewrite the integral:** $$\int \left(1 + \sqrt{a} \cdot x^{-\frac{1}{2}}\right) dx = \int 1 \, dx + \sqrt{a} \int x^{-\frac{1}{2}} \, dx$$ 4. **Integrate each term:** - Integral of 1 with respect to $x$ is $x$. - Integral of $x^{-\frac{1}{2}}$ is $$\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}}$$ 5. **Combine results:** $$x + \sqrt{a} \cdot 2x^{\frac{1}{2}} + C = x + 2\sqrt{a} \sqrt{x} + C$$ 6. **Final answer:** $$\int \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x}} \, dx = x + 2\sqrt{a} \sqrt{x} + C$$