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. **Rewrite the integrand:** Split the fraction into two parts: $$\frac{\sqrt{x}}{\sqrt{x}} + \frac{\sqrt{a}}{\sqrt{x}} = 1 + \sqrt{a} x^{-\frac{1}{2}}$$ 3. **Integral formula:** Recall that $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$. 4. **Integrate term-by-term:** $$\int \left(1 + \sqrt{a} x^{-\frac{1}{2}}\right) dx = \int 1 \, dx + \sqrt{a} \int x^{-\frac{1}{2}} \, dx$$ 5. **Calculate each integral:** - $$\int 1 \, dx = x + C_1$$ - $$\int x^{-\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C_2 = 2 \sqrt{x} + C_2$$ 6. **Combine results:** $$x + \sqrt{a} \cdot 2 \sqrt{x} + C = x + 2 \sqrt{a} \sqrt{x} + C$$ 7. **Final answer:** $$\int \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}} \, dx = x + 2 \sqrt{a} \sqrt{x} + C$$ This completes the integration.