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28. [0/1 Points] DETAILS MY NOTES SCALCET8 4.9.023.
Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of and ,

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Ask by Ruiz King. in the United States

Jan 29,2025

Question

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28. [0/1 Points] DETAILS MY NOTES SCALCET8 4.9.023.
Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of and ,

Need Help? Wateh II
SUBMIT ANSWER

Ask by Ruiz King. in the United States

Jan 29,2025

UpStudy AI · Accepted Answer

To find the antiderivative $$ F $$ of the function $$ f $$ given by $$ F(x) = 5x^4 - 2x^5 $$ and the condition $$ F(0) = 4 $$, we will follow these steps:

1. **Find the general antiderivative**: The function $$ f(x) $$ is given as $$ f(x) = 5x^4 - 2x^5 $$. To find the antiderivative, we will integrate $$ f(x) $$.

2. **Integrate $$ f(x) $$**:
   $$
   F(x) = \int (5x^4 - 2x^5) \, dx
   $$

3. **Calculate the integral**:
   - The integral of $$ 5x^4 $$ is $$ x^5 $$.
   - The integral of $$ -2x^5 $$ is $$ -\frac{2}{6}x^6 = -\frac{1}{3}x^6 $$.

Therefore, we have:
   $$
   F(x) = x^5 - \frac{1}{3}x^6 + C
   $$
   where $$ C $$ is the constant of integration.

4. **Use the condition $$ F(0) = 4 $$** to find $$ C $$:
   $$
   F(0) = 0^5 - \frac{1}{3}(0^6) + C = C
   $$
   Setting this equal to 4 gives:
   $$
   C = 4
   $$

5. **Write the final expression for $$ F(x) $$**:
   $$
   F(x) = x^5 - \frac{1}{3}x^6 + 4
   $$

Thus, the antiderivative $$ F $$ that satisfies the given condition is:
$$
F(x) = x^5 - \frac{1}{3}x^6 + 4
$$

Now, let's summarize the final answer:
$$
F(x) = x^5 - \frac{1}{3}x^6 + 4
$$
$$ F(x) = x^5 - \frac{1}{3}x^6 + 4 $$

SUBMIT ANSWER
28. [0/1 Points] DETAILS MY NOTES SCALCET8 4.9.023.
Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of and ,

Need Help? Wateh II
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SUBMIT ANSWER 28. [0/1 Points] DETAILS MY NOTES SCALCET8 4.9.023. Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of and , Need Help? Wateh II SUBMIT ANSWER

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28. [0/1 Points] DETAILS MY NOTES SCALCET8 4.9.023.
Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of and ,

Need Help? Wateh II
SUBMIT ANSWER