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The given problem asks to evaluate the indefinite integral:

\[
\int \left( \frac{3\sqrt{x} + 2}{\sqrt{x}} \right) dx
\]

### Step-by-step Solution:

1. **Simplify the integrand**:
   Break the integrand into two separate terms:

   \[
   \frac{3\sqrt{x}}{\sqrt{x}} + \frac{2}{\sqrt{x}} = 3 + \frac{2}{\sqrt{x}}
   \]

   Since \(\frac{3\sqrt{x}}{\sqrt{x}} = 3\) and \(\frac{2}{\sqrt{x}}\) can be rewritten as \(2x^{-\frac{1}{2}}\), the expression simplifies to:

   \[
   3 + 2x^{-\frac{1}{2}}
   \]

2. **Integrate term by term**:
   Now, we can integrate each term separately.

   - The integral of \(3\) is:

     \[
     \int 3 dx = 3x
     \]

   - The integral of \(2x^{-\frac{1}{2}}\) is:

     \[
     \int 2x^{-\frac{1}{2}} dx = 2 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 4x^{\frac{1}{2}}
     \]

3. **Final answer**:
   Combining both results:

   \[
   3x + 4\sqrt{x} + C
   \]

   Since the problem specifies not to write the arbitrary constant \(C\), the final result is:

   \[
   3x + 4\sqrt{x}
   \]

Would you like more details or clarifications on any part of the solution?

Here are 5 related questions for further practice:
1. How would you evaluate \(\int \frac{1}{\sqrt{x}} dx\)?
2. What is the indefinite integral of \(x^n\) for any real \(n\)?
3. How would you find \(\int \left( 5x^{3} + 2x^{-1} \right) dx\)?
4. How do you apply the power rule of integration to negative exponents?
5. What would change in the evaluation if the limits of integration were from 1 to 4?

**Tip**: When integrating, always simplify the integrand as much as possible before applying basic rules.

Evaluate the indefinite integral \(\int \left( \frac{3\sqrt{x} + 2}{\sqrt{x}} \right) dx\).

Learn how to evaluate the indefinite integral \(\int \left( \frac{3\sqrt{x} + 2}{\sqrt{x}} \right) dx\) with a detailed step-by-step solution. This problem covers algebraic simplification and the power rule of integration. Suitable for students in Grades 11-12.

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