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To find the antiderivative of the expression \( \sqrt{x} + \frac{1}{\sqrt{x}} \), we can rewrite it in terms of exponents and integrate term-by-term. Here's the solution:

### Step-by-Step Solution
1. **Rewrite the terms** using exponents:
   \[
   \sqrt{x} = x^{\frac{1}{2}} \quad \text{and} \quad \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}.
   \]
   So the expression becomes:
   \[
   x^{\frac{1}{2}} + x^{-\frac{1}{2}}.
   \]

2. **Set up the integral**:
   \[
   \int \left( x^{\frac{1}{2}} + x^{-\frac{1}{2}} \right) \, dx.
   \]

3. **Integrate each term separately**:
   - For \( x^{\frac{1}{2}} \):
     \[
     \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} x^{\frac{3}{2}}.
     \]
   - For \( x^{-\frac{1}{2}} \):
     \[
     \int x^{-\frac{1}{2}} \, dx = \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 x^{\frac{1}{2}}.
     \]

4. **Combine the results**:
   \[
   \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C.
   \]

Thus, the antiderivative is:
\[
\frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C.
\]

### Final Answer
The correct answer should match this form, which may correspond to one of the given choices in your options. If this matches with option **(A)**, that would be your answer.

Would you like further details or have any questions?

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1. How do you find the antiderivative of more complex functions?
2. Why is the constant \( C \) important in indefinite integrals?
3. How can fractional exponents be interpreted in terms of radicals?
4. How do you verify if an antiderivative is correct?
5. What are common mistakes when integrating terms with fractional powers?

**Tip:** Always rewrite radicals as exponents when integrating; it simplifies the process significantly.

The antiderivative of \( \sqrt{x} + \frac{1}{\sqrt{x}} \)

Explore the step-by-step process to find the antiderivative of \( \sqrt{x} + \frac{1}{\sqrt{x}} \). This solution involves rewriting radicals as exponents and applying the Power Rule for Integration, suitable for high school students in Grades 10-12.

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