The problem involves evaluating the integral $\int (x + 11)^3 dx$ using the substitution rule. Let's verify the correctness of the solution step by step.

Given:

$\int (x + 11)^3 dx$

Step-by-step verification:

1. Substitution: Let $u = x + 11$, which implies $du = dx$.
   The substitution appears to be correct.

2. Changing the integral: Substituting $u = x + 11$, we get: $\int (x + 11)^3 dx = \int u^3 du$ This transformation is valid.

3. Integration of $u^3$: The integral of $u^3$ is: $\int u^3 du = \frac{u^4}{4} + C$ This step is correct.

4. Back-substitution: Since $u = x + 11$, replacing $u$ gives: $\frac{u^4}{4} + C = \frac{(x + 11)^4}{4} + C$ This is correctly done.

Correct Answer:

The solution and its steps are valid. Thus, the correct statement is: $\text{The illustration is correct because all parts are correct.}$

Do you want further clarification, or have any questions?

Here are 5 related questions you might explore:

1. How is the substitution rule applied to definite integrals?
2. Why is the substitution $u = x + 11$ helpful in this problem?
3. Can substitution methods handle integrals where $du \neq dx$?
4. How would the answer change if the power were $(x + 11)^n$?
5. What are the common errors students make when using substitution?

Tip: Always verify the substitution and back-substitution steps carefully to avoid mistakes in your integration.