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To find the antiderivative \( F(x) \) of \( f(x) = 3x^3 - 5x^2 + 9x - 10 \), we need to integrate each term individually. 

### Step-by-Step Solution:

1. **Integrate each term of \( f(x) \):**
   \[
   \int 3x^3 \, dx = \frac{3x^4}{4} = \frac{3}{4}x^4
   \]
   \[
   \int -5x^2 \, dx = -\frac{5x^3}{3} = -\frac{5}{3}x^3
   \]
   \[
   \int 9x \, dx = \frac{9x^2}{2} = \frac{9}{2}x^2
   \]
   \[
   \int -10 \, dx = -10x
   \]

2. **Combine the terms to write the general antiderivative \( F(x) \):**
   \[
   F(x) = \frac{3}{4}x^4 - \frac{5}{3}x^3 + \frac{9}{2}x^2 - 10x + C
   \]
   where \( C \) is the constant of integration.

3. **Identify coefficients for the answer fields:**
   - \( A = \frac{3}{4} \)
   - \( B = -\frac{5}{3} \)
   - \( C = \frac{9}{2} \)
   - \( D = -10 \)

### Final Answer:
- **A is** \( \frac{3}{4} \)
- **B is** \( -\frac{5}{3} \)
- **C is** \( \frac{9}{2} \)
- **D is** \( -10 \)

Would you like more details, or do you have any questions?

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1. How would the antiderivative change if \( f(x) \) had higher powers?
2. What happens to \( F(x) \) if you add or subtract a constant to \( f(x) \)?
3. How can you check if your antiderivative is correct?
4. What is the physical meaning of an antiderivative in applied problems?
5. How does the choice of constant \( C \) affect the graph of \( F(x) \)?

**Tip:** Remember, the antiderivative is unique up to a constant \( C \), which represents the family of solutions.

Consider the function f(x) = 3x^3 - 5x^2 + 9x - 10. An antiderivative of f(x) is F(x) = Ax^4 + Bx^3 + Cx^2 + Dx where A is __ and B is __ and C is __ and D is __.

Learn how to find the antiderivative of the polynomial function f(x) = 3x^3 - 5x^2 + 9x - 10. This step-by-step guide explains how to integrate each term, resulting in F(x) = Ax^4 + Bx^3 + Cx^2 + Dx. Ideal for high school students in Grades 11-12.

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