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?

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.