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The given problem involves evaluating the integral:

\[
\int x^4 (x^5 - 9)^3 \, dx, \quad \text{with the substitution } u = x^5 - 9
\]

Let's break down the steps for solving it using this substitution.

### Step-by-Step Solution:

1. **Substitution**:
   \[
   u = x^5 - 9
   \]
   Therefore, the derivative of \( u \) with respect to \( x \) is:
   \[
   \frac{du}{dx} = 5x^4 \quad \text{or} \quad du = 5x^4 dx
   \]
   Now, we can rewrite \( x^4 dx \) as:
   \[
   x^4 dx = \frac{1}{5} du
   \]

2. **Rewriting the Integral**:
   Substituting into the original integral:
   \[
   \int x^4 (x^5 - 9)^3 \, dx = \int (u^3) \left( \frac{1}{5} du \right)
   \]
   This simplifies to:
   \[
   \frac{1}{5} \int u^3 du
   \]

3. **Integrating**:
   Now, integrate \( u^3 \):
   \[
   \frac{1}{5} \cdot \frac{u^4}{4} + C = \frac{1}{20} u^4 + C
   \]

4. **Substitute Back**:
   Replace \( u \) with \( x^5 - 9 \):
   \[
   \frac{1}{20} (x^5 - 9)^4 + C
   \]

### Final Answer:
The correct answer is:

\[
\boxed{A: \frac{1}{20}(x^5 - 9)^4 + C}
\]

Would you like further details on any step? Here are some additional related questions to consider:

1. How would you handle substitution for integrals with trigonometric functions?
2. What happens if we integrate a similar expression but with higher powers of \( x \)?
3. How can we approach definite integrals using substitution?
4. Can we verify the solution by differentiating the result?
5. What are the conditions for substitution to be a useful method for solving integrals?

**Tip:** When using substitution for integration, always ensure that the differential is correctly replaced with the new variable to avoid errors in the final expression.

Evaluate the integral using the given substitution: ∫x^4(x^5 - 9)^3 dx, u = x^5 - 9

This problem demonstrates how to evaluate the integral of x^4(x^5 - 9)^3 using substitution. It covers essential calculus concepts such as the substitution method and power rule. Suitable for undergraduate calculus students.

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