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The problem asks us to solve for \( f(64) \) given the equation:

\[
\int_4^{x^3} f(t) \, dt = \ln(9x^3) + x^4
\]

We are specifically tasked with finding \( f(64) \), so we need to use this relation to first find a general expression for \( f(t) \) and then evaluate it when \( t = 64 \).

### Step-by-Step Solution:

1. **Differentiate both sides of the equation with respect to \( x \)** using the Fundamental Theorem of Calculus and the chain rule.

   \[
   \frac{d}{dx} \left( \int_4^{x^3} f(t) \, dt \right) = \frac{d}{dx} \left( \ln(9x^3) + x^4 \right)
   \]

   The left-hand side differentiates to:

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

   The right-hand side differentiates to:

   \[
   \frac{d}{dx} \ln(9x^3) = \frac{1}{9x^3} \cdot 27x^2 = \frac{3}{x}, \quad \frac{d}{dx} x^4 = 4x^3
   \]

   So, we now have the equation:

   \[
   3x^2 f(x^3) = \frac{3}{x} + 4x^3
   \]

2. **Solve for \( f(x^3) \):**

   Divide both sides of the equation by \( 3x^2 \):

   \[
   f(x^3) = \frac{\frac{3}{x} + 4x^3}{3x^2}
   \]

   Simplify the terms:

   \[
   f(x^3) = \frac{1}{x^3} + \frac{4}{3}x
   \]

3. **Substitute \( x = 4 \) into \( f(x^3) \):**

   To find \( f(64) \), we observe that \( x^3 = 64 \) when \( x = 4 \). Therefore, we substitute \( x = 4 \) into the expression for \( f(x^3) \):

   \[
   f(64) = \frac{1}{4^3} + \frac{4}{3} \cdot 4
   \]

   Calculate the terms:

   \[
   f(64) = \frac{1}{64} + \frac{16}{3}
   \]

4. **Simplify the expression:**

   Convert the terms to have a common denominator:

   \[
   f(64) = \frac{1}{64} + \frac{16 \times 64}{3 \times 64} = \frac{1}{64} + \frac{1024}{192}
   \]

   Therefore:

   \[
   f(64) = \frac{1}{64} + \frac{16}{3}
   \]

   This gives the final value for \( f(64) \). Would you like further clarification on any part of this process, or the final calculation?

### Relative Questions:
1. How does the Fundamental Theorem of Calculus apply in this scenario?
2. What are the key steps in using the chain rule during differentiation?
3. Why do we differentiate both sides of the given integral equation?
4. What is the significance of substituting \( x = 4 \) to find \( f(64) \)?
5. How do you handle combining fractions with different denominators in this context?

### Tip:
Always remember to apply both the Fundamental Theorem of Calculus and the chain rule carefully when differentiating equations that involve integrals with variable limits.

Considere como válida la siguiente relación: ∫ f(t) dt = ln(9x³) + x⁴. El valor de f(64) es:

This problem requires solving for f(64) using integral calculus and differentiation. Apply the Fundamental Theorem of Calculus and chain rule to differentiate both sides of the equation and solve. Suitable for undergraduate calculus students.

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