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We are tasked with finding the most general antiderivative of the given function: 

\[
\int (2x^3 - 7x + 4) \, dx
\]

### Step-by-step Solution:

1. **Apply the Power Rule for Integration:**
   The general power rule for integration states:
   \[
   \int x^n \, dx = \frac{x^{n+1}}{n+1} + C
   \]
   where \(C\) is the constant of integration.

2. **Integrating Term by Term:**

   - For \(2x^3\):
     \[
     \int 2x^3 \, dx = 2 \cdot \frac{x^4}{4} = \frac{x^4}{2}
     \]

   - For \(-7x\):
     \[
     \int -7x \, dx = -7 \cdot \frac{x^2}{2} = -\frac{7x^2}{2}
     \]

   - For the constant \(4\):
     \[
     \int 4 \, dx = 4x
     \]

3. **Combining the Results:**
   The general antiderivative is:
   \[
   \frac{x^4}{2} - \frac{7x^2}{2} + 4x + C
   \]

### Matching with the Given Options:
The correct answer is option **D**:
\[
\frac{1}{2}x^4 - \frac{7}{2}x^2 + 4x + C
\]

Let me know if you'd like more details, or if you have any questions!

---

Here are 5 follow-up questions to expand on this:

1. How would you integrate a function like \(3x^5 - 2x^2 + 1\)?
2. What happens if there is a coefficient of \(\frac{1}{x}\) in the integrand?
3. Can the power rule for integration be used on trigonometric functions?
4. What is the importance of the constant \(C\) in an indefinite integral?
5. How would you check if your antiderivative is correct?

**Tip:** When integrating, always ensure that the exponent increases by 1 and divide by the new exponent—this is key to mastering the power rule!

Find the most general antiderivative of \( 2x^3 - 7x + 4 \).

Learn how to find the most general antiderivative of the function \( 2x^3 - 7x + 4 \) using the power rule for integration. Detailed step-by-step solution included. Suitable for high school and early college students.

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