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To solve the given problem, we need to find the most general antiderivative of the polynomial expression:

\[
\int (40x^3 + 12x^2 - 9x + 14) \, dx
\]

### Steps:

1. **Integrate each term separately** using the power rule for integration:
   \[
   \int x^n \, dx = \frac{x^{n+1}}{n+1} + C
   \]

2. **Apply the rule** to each term:
   - \(\int 40x^3 \, dx = 40 \cdot \frac{x^4}{4} = 10x^4\)
   - \(\int 12x^2 \, dx = 12 \cdot \frac{x^3}{3} = 4x^3\)
   - \(\int -9x \, dx = -9 \cdot \frac{x^2}{2} = -\frac{9x^2}{2}\)
   - \(\int 14 \, dx = 14x\)

3. **Add the constant of integration \(C\)** to account for the most general antiderivative.

### Final result:
\[
10x^4 + 4x^3 - \frac{9x^2}{2} + 14x + C
\]

Let me know if you'd like further clarification or details!

### Related questions:
1. How does the power rule for integration work for negative exponents?
2. What happens when integrating a constant like 14?
3. Why do we add a constant of integration when finding antiderivatives?
4. Can you explain how to integrate polynomials that involve fractional powers?
5. How does this approach change for definite integrals?

**Tip:** When integrating, always check that you properly adjust the exponent of each variable term and don't forget to divide by the new exponent.

Find the most general antiderivative of the polynomial expression: ∫(40x^3 + 12x^2 - 9x + 14) dx

Learn how to find the most general antiderivative of the polynomial expression 40x^3 + 12x^2 - 9x + 14 with detailed steps using the power rule for integration. Suitable for high school and college calculus students.

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