which antiderivative of f(x)=x*sqrt(x^2+4) gets the value 9 when x=sqrt(5)

To find the antiderivative \( F(x) \) of the function \( f(x) = x\sqrt{x^2 + 4} \) that satisfies \( F(\sqrt{5}) = 9 \), let's proceed with the following steps:

### Step 1: Find the Antiderivative
We want to find \( F(x) \) where:
\[
F'(x) = f(x) = x\sqrt{x^2 + 4}
\]

To solve this, we can use the substitution method.

Let:
\[
u = x^2 + 4
\]
Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = 2x \quad \text{or} \quad du = 2x \, dx
\]

Now, rewrite the integral:
\[
\int x\sqrt{x^2 + 4} \, dx = \int \sqrt{u} \cdot \frac{du}{2} = \frac{1}{2} \int u^{1/2} \, du
\]

The antiderivative of \( u^{1/2} \) is:
\[
\frac{1}{2} \int u^{1/2} \, du = \frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{3} u^{3/2} + C
\]

Substituting \( u = x^2 + 4 \) back into the expression:
\[
F(x) = \frac{1}{3} (x^2 + 4)^{3/2} + C
\]

### Step 2: Use the Initial Condition to Find \( C \)
We are given that \( F(\sqrt{5}) = 9 \). Substituting \( x = \sqrt{5} \) into the antiderivative:
\[
9 = \frac{1}{3} ((\sqrt{5})^2 + 4)^{3/2} + C
\]
\[
9 = \frac{1}{3} (5 + 4)^{3/2} + C
\]
\[
9 = \frac{1}{3} \cdot 9^{3/2} + C
\]
\[
9 = \frac{1}{3} \cdot 27 + C
\]
\[
9 = 9 + C
\]
\[
C = 0
\]

### Final Answer
Thus, the antiderivative that satisfies the condition is:
\[
F(x) = \frac{1}{3} (x^2 + 4)^{3/2}
\]

Would you like to explore further details or ask any questions?

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Here are 5 related questions you might be interested in:
1. How would the process change if the function were \( f(x) = x\sqrt{x^2 - 4} \)?
2. What is the physical interpretation of the constant \( C \) in antiderivatives?
3. How do you find the antiderivative if the integral involves a trigonometric function?
4. What are other methods for solving integrals besides substitution?
5. How can we verify the solution of an antiderivative problem?

**Tip:** Always double-check your substitution, especially when transforming back to the original variable, to ensure consistency.

Learn how to find the antiderivative of the function x*sqrt(x^2 + 4) that satisfies F(sqrt(5)) = 9. Detailed step-by-step solution using the substitution method in integration. Suitable for advanced high school students and beyond.

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