To solve the integral of \( 3x \) with respect to \( x \), we need to apply the basic rules of integration. Here's the step-by-step process:

1. **Identify the function to be integrated**:
   \[
   \int 3x \, dx
   \]

2. **Apply the constant multiple rule**:
   The integral of a constant times a function is the constant times the integral of the function. So we can factor out the constant 3:
   \[
   3 \int x \, dx
   \]

3. **Integrate the function**:
   Use the power rule of integration, which states that:
   \[
   \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for} \quad n \neq -1
   \]
   In this case, \( n = 1 \), so:
   \[
   \int x \, dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C
   \]

4. **Combine the results**:
   Multiply the result of the integral by the constant 3:
   \[
   3 \left( \frac{x^2}{2} + C \right) = \frac{3x^2}{2} + 3C
   \]

Since \( 3C \) is also a constant, we can simply denote it as \( C \). Therefore, the final result is:
\[
\int 3x \, dx = \frac{3x^2}{2} + C
\]

Would you like any further details or explanations?

Here are some related questions you might find interesting:

1. How do you integrate \( x^2 \)?
2. What is the integral of \( 5x^3 \)?
3. How do you handle integrals involving trigonometric functions?
4. What are the steps to integrate a polynomial function?
5. How do you find the definite integral of \( 3x \) from 1 to 4?
6. What is the significance of the constant of integration, \( C \)?
7. How do you apply the power rule for integration to functions with negative exponents?
8. What is the process for integrating a composite function?

**Tip**: When integrating, always remember to add the constant of integration, \( C \), to account for all possible antiderivatives.

Learn how to integrate the function 3x with respect to x using the constant multiple rule and power rule of integration. This problem covers basic calculus concepts and is suitable for students in Grades 11-12.

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