For the following function​ f, find the antiderivative F that satisfies the given condition.

​f(x)equals=9 StartRoot x EndRoot plus 39x+3​; 

​F(11​)equals=88

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Part 1

The antiderivative that satisfies the given condition is 

​F(x)equals=enter your response here.

To find the antiderivative \( F(x) \) of the function \( f(x) = 9\sqrt{x} + 39x + 3 \) that satisfies the condition \( F(11) = 88 \), we will follow these steps:

1. Find the general antiderivative \( F(x) \) of \( f(x) \).
2. Use the given condition \( F(11) = 88 \) to determine the constant of integration \( C \).

### Step 1: Find the General Antiderivative

First, we will find the antiderivative of each term in \( f(x) \).

\[ f(x) = 9\sqrt{x} + 39x + 3 \]

**Antiderivative of \( 9\sqrt{x} \)**:

Recall that \( \sqrt{x} = x^{1/2} \).

\[ \int 9\sqrt{x} \, dx = \int 9x^{1/2} \, dx \]

Using the power rule for integration \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \):

\[ \int 9x^{1/2} \, dx = 9 \int x^{1/2} \, dx = 9 \left( \frac{x^{3/2}}{3/2} \right) = 9 \left( \frac{2}{3} x^{3/2} \right) = 6x^{3/2} \]

**Antiderivative of \( 39x \)**:

\[ \int 39x \, dx = 39 \int x \, dx = 39 \left( \frac{x^2}{2} \right) = \frac{39}{2} x^2 \]

**Antiderivative of \( 3 \)**:

\[ \int 3 \, dx = 3x \]

Combining these results, the general antiderivative is:

\[ F(x) = 6x^{3/2} + \frac{39}{2} x^2 + 3x + C \]

### Step 2: Use the Given Condition

We use the condition \( F(11) = 88 \) to find \( C \):

\[ F(11) = 6(11)^{3/2} + \frac{39}{2}(11)^2 + 3(11) + C = 88 \]

Calculate each term separately:

- \( 11^{3/2} = \sqrt{11^3} = \sqrt{1331} \approx 36.4829 \)
- \( 6 \times 36.4829 \approx 218.8974 \)
- \( 11^2 = 121 \)
- \( \frac{39}{2} \times 121 = 2359.5 \)
- \( 3 \times 11 = 33 \)

Substitute these into the equation:

\[ 218.8974 + 2359.5 + 33 + C = 88 \]

Combine the constants:

\[ 2611.3974 + C = 88 \]

Solve for \( C \):

\[ C = 88 - 2611.3974 \approx -2523.3974 \]

So the antiderivative \( F(x) \) that satisfies the condition is:

\[ F(x) = 6x^{3/2} + \frac{39}{2} x^2 + 3x - 2523.3974 \]

Would you like details or have any questions?

Here are some related questions you might find interesting:

1. What is the power rule for integration?
2. How do you handle definite integrals with given limits?
3. Can you find the antiderivative of a trigonometric function?
4. How do you verify an antiderivative using differentiation?
5. What are common techniques for finding antiderivatives of complex functions?
6. How do you solve initial value problems in calculus?
7. Can antiderivatives be applied to real-world problems?
8. How do constants of integration affect the graph of an antiderivative?

**Tip:** Always double-check your integration steps and calculations, especially when handling constants of integration and specific conditions given in problems.

Learn how to find the antiderivative of the function 9√x + 39x + 3 that satisfies the condition F(11) = 88. This problem covers integration techniques, including the power rule, and solving for a constant of integration using a specific condition. Suitable for students in Grades 11-12.

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