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

​f(x)equals=3 StartRoot x EndRoot plus 93x+9​; 

​F(11​)equals=66




To find the antiderivative \( F(x) \) of the function \( f(x) = 3\sqrt{x} + 9x + 9 \) that satisfies the condition \( F(11) = 66 \), follow these steps:

1. **Find the indefinite integral of \( f(x) \):**
   \[
   F(x) = \int f(x) \, dx = \int \left( 3\sqrt{x} + 9x + 9 \right) \, dx
   \]

2. **Integrate each term separately:**

   - For \( 3\sqrt{x} \):
     \[
     3\sqrt{x} = 3x^{1/2} \implies \int 3x^{1/2} \, dx = 3 \cdot \frac{x^{3/2}}{3/2} = 2x^{3/2}
     \]

   - For \( 9x \):
     \[
     9x \implies \int 9x \, dx = 9 \cdot \frac{x^2}{2} = \frac{9}{2}x^2
     \]

   - For \( 9 \):
     \[
     9 \implies \int 9 \, dx = 9x
     \]

3. **Combine the results of the integrations:**
   \[
   F(x) = 2x^{3/2} + \frac{9}{2}x^2 + 9x + C
   \]
   where \( C \) is the constant of integration.

4. **Use the initial condition \( F(11) = 66 \) to find \( C \):**
   \[
   F(11) = 2(11)^{3/2} + \frac{9}{2}(11)^2 + 9(11) + C = 66
   \]

   Calculate each term separately:
   \[
   2(11)^{3/2} = 2 \cdot \sqrt{11^3} = 2 \cdot \sqrt{1331} \approx 2 \cdot 36.4829 = 72.9658
   \]

   \[
   \frac{9}{2}(11)^2 = \frac{9}{2} \cdot 121 = 4.5 \cdot 121 = 544.5
   \]

   \[
   9(11) = 99
   \]

   Now substitute these values into the equation:
   \[
   72.9658 + 544.5 + 99 + C = 66
   \]

   Combine the constants and solve for \( C \):
   \[
   716.4658 + C = 66 \implies C = 66 - 716.4658 = -650.4658
   \]

5. **Write the final antiderivative:**
   \[
   F(x) = 2x^{3/2} + \frac{9}{2}x^2 + 9x - 650.4658
   \]

So, the antiderivative \( F(x) \) that satisfies the given condition is:
\[
F(x) = 2x^{3/2} + \frac{9}{2}x^2 + 9x - 650.4658
\]

Would you like more details or have any questions? Here are some related questions to further explore the topic:

1. How do you find the antiderivative of a composite function?
2. What are the properties of definite and indefinite integrals?
3. How do you verify the correctness of an antiderivative?
4. What are some common techniques for integrating complex functions?
5. Can you explain the Fundamental Theorem of Calculus?
6. How do you handle integration by parts?
7. What are some applications of antiderivatives in real-world problems?
8. How do you find the area under a curve using antiderivatives?

**Tip:** Always check your integration by differentiating your result to ensure you recover the original function.

Learn how to find the antiderivative F(x) of the function f(x) = 3√x + 93x + 9 that satisfies the condition F(11) = 66. Step-by-step solution involving integration techniques and the Fundamental Theorem of Calculus. Suitable for advanced high school and college students.

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