To evaluate the integral

$\int x^4 (x^5 - 9)^3 \, dx$

using the given substitution $u = x^5 - 9$, we can follow these steps:

1. Differentiate the substitution: $u = x^5 - 9 \Rightarrow \frac{du}{dx} = 5x^4 \Rightarrow dx = \frac{du}{5x^4}$

2. Rewrite the integral in terms of $u$: Substitute $x^4$ and $dx$ into the integral: $\int x^4 (x^5 - 9)^3 \, dx = \int x^4 \cdot u^3 \cdot \frac{du}{5x^4}$

3. Simplify: The $x^4$ terms cancel out, so we are left with: $\int u^3 \cdot \frac{1}{5} \, du = \frac{1}{5} \int u^3 \, du$

4. Integrate with respect to $u$: $\frac{1}{5} \int u^3 \, du = \frac{1}{5} \cdot \frac{u^4}{4} = \frac{u^4}{20}$

5. Substitute back $u = x^5 - 9$: $\frac{(x^5 - 9)^4}{20} + C$

So, the correct answer is:

$\boxed{\frac{(x^5 - 9)^4}{20} + C}$

This corresponds to option C.

Would you like further details or have any questions?

Related Questions

1. How does substitution simplify integration in certain integrals?
2. What is the difference between indefinite and definite integration?
3. How can one identify the best substitution to use in integrals?
4. What are some common techniques for integrating polynomial functions?
5. How does the chain rule relate to substitution in integrals?

Tip

Always double-check that the differential $dx$ is correctly converted in terms of $du$ when performing substitutions in integration.