To solve the integral:

\[
\int 3\sqrt{x} \, dx
\]

### Step 1: Rewrite the integrand
We can express the square root as a fractional exponent:
\[
\sqrt{x} = x^{1/2}
\]
Thus, the integral becomes:
\[
\int 3x^{1/2} \, dx
\]

### Step 2: Apply the power rule
The power rule for integration states that:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where \( n \neq -1 \))}
\]
In this case, \( n = \frac{1}{2} \). Applying the rule:

\[
\int 3x^{1/2} \, dx = 3 \cdot \frac{x^{1/2 + 1}}{1/2 + 1} + C
\]
\[
= 3 \cdot \frac{x^{3/2}}{3/2} + C
\]

### Step 3: Simplify
To simplify the fraction:
\[
3 \cdot \frac{x^{3/2}}{3/2} = 3 \cdot \frac{2}{3} \cdot x^{3/2} = 2x^{3/2}
\]

### Final answer:
\[
\int 3\sqrt{x} \, dx = 2x^{3/2} + C
\]

Would you like further explanation or additional steps?

Here are 5 related questions:
1. How do you solve integrals with negative exponents?
2. What is the integral of \( \frac{1}{x} \)?
3. How do you handle integrals with trigonometric functions?
4. How do definite integrals differ from indefinite integrals?
5. Can you explain the substitution method for integrals?

**Tip**: When dealing with powers, always remember to adjust the exponent and divide by the new exponent when integrating!

Learn how to integrate the function 3√x with a detailed step-by-step solution. This guide covers key calculus concepts such as integration and the power rule for integration, making it suitable for students in Grades 11-12. Enhance your understanding of indefinite integrals with this comprehensive explanation.

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