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.