To find the values of $A$, $B$, $C$, and $D$ for the antiderivative $F(x)$ of the function $f(x) = 10x^3 - 5x^2 + 3x - 5$, we need to integrate $f(x)$ term-by-term:

Step-by-Step Solution

1. Integrate $10x^3$: $\int 10x^3 \, dx = \frac{10}{4}x^4 = 2.5x^4$ So, $A = 2.5$.

2. Integrate $-5x^2$: $\int -5x^2 \, dx = \frac{-5}{3}x^3 = -\frac{5}{3}x^3$ So, $B = -\frac{5}{3}$.

3. Integrate $3x$: $\int 3x \, dx = \frac{3}{2}x^2 = 1.5x^2$ So, $C = 1.5$.

4. Integrate $-5$: $\int -5 \, dx = -5x$ So, $D = -5$.

Final Answer

The values are:

• $A = 2.5$
• $B = -\frac{5}{3}$
• $C = 1.5$
• $D = -5$

The antiderivative $F(x)$ is therefore: $F(x) = 2.5x^4 - \frac{5}{3}x^3 + 1.5x^2 - 5x$

Would you like further details on the integration steps or have any other questions?

Here are some related questions for practice:

1. How would you find the antiderivative if $C \neq 0$?
2. What happens to the constants if the function $f(x)$ included higher powers of $x$?
3. How would you confirm that $F(x)$ is correct by differentiation?
4. What is the geometric interpretation of the antiderivative in terms of area under $f(x)$?
5. How does the degree of $f(x)$ affect the degree of $F(x)$?

Tip: Always double-check constants by differentiating your antiderivative to ensure it matches the original function.