Dada la función 𝐹, determine la función 𝑓 tal que ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶
c) ∫ 𝑓(𝑥)𝑑𝑥 = 𝑥
3𝑥 + �

To determine the function \( f(x) \) such that the integral of \( f(x) \) equals the given function \( F(x) \), we need to differentiate the provided function \( F(x) \) with respect to \( x \).

Given:
\[
\int f(x) \, dx = F(x) + C
\]

**Part (c)** is:
\[
\int f(x) \, dx = \frac{x^2}{3x + C}
\]

To find \( f(x) \), we differentiate both sides with respect to \( x \):

\[
f(x) = \frac{d}{dx}\left(\frac{x^2}{3x + C}\right)
\]

We'll apply the quotient rule for differentiation:

\[
f(x) = \frac{(3x + C) \cdot \frac{d}{dx}(x^2) - x^2 \cdot \frac{d}{dx}(3x + C)}{(3x + C)^2}
\]

Differentiating the terms:

\[
\frac{d}{dx}(x^2) = 2x \quad \text{and} \quad \frac{d}{dx}(3x + C) = 3
\]

So:

\[
f(x) = \frac{(3x + C) \cdot 2x - x^2 \cdot 3}{(3x + C)^2}
\]

Simplifying:

\[
f(x) = \frac{6x^2 + 2Cx - 3x^2}{(3x + C)^2} = \frac{3x^2 + 2Cx}{(3x + C)^2}
\]

Thus, the function \( f(x) \) that satisfies the given integral equation is:

\[
f(x) = \frac{3x^2 + 2Cx}{(3x + C)^2}
\]

Would you like further details or have any questions about this process?

Here are some related questions to explore:
1. How does the quotient rule for differentiation work?
2. What is the significance of the constant \( C \) in indefinite integrals?
3. Can you explain how to simplify expressions after applying the quotient rule?
4. What are some common mistakes to avoid when applying the quotient rule?
5. How would the problem change if \( C \) were a specific constant, like 0?

**Tip:** Always check your differentiation by substituting back into the original function to ensure that the integral equation holds.

Learn how to determine the function f(x) from the integral equation ∫ f(x) dx = F(x) + C, with a step-by-step solution provided. This problem involves advanced mathematical concepts such as integration, differentiation, and the quotient rule. Suitable for advanced mathematics students.

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