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We are tasked with evaluating the following integral:

\[
\int \frac{18 \tan^2 x \sec^2 x}{(6 + \tan^3 x)^2} \, dx
\]

### Step-by-step Solution

1. **Substitution**: 
   Since we have a composition involving \( \tan^3 x \), we attempt the substitution:
   \[
   u = 6 + \tan^3 x
   \]
   Differentiating both sides with respect to \( x \):
   \[
   \frac{du}{dx} = 3 \tan^2 x \sec^2 x
   \]
   Solving for \( dx \), we get:
   \[
   dx = \frac{du}{3 \tan^2 x \sec^2 x}
   \]

2. **Simplify the Integral**: 
   Substituting into the integral, the expression becomes:
   \[
   \int \frac{18 \tan^2 x \sec^2 x}{u^2} \cdot \frac{du}{3 \tan^2 x \sec^2 x}
   \]
   Notice that \( \tan^2 x \sec^2 x \) cancels out, leaving:
   \[
   \int \frac{18}{3 u^2} \, du
   \]
   Simplifying:
   \[
   \int \frac{6}{u^2} \, du
   \]

3. **Integrating**: 
   The integral of \( u^{-2} \) is:
   \[
   \int u^{-2} \, du = -\frac{1}{u}
   \]
   Therefore:
   \[
   \int \frac{6}{u^2} \, du = -\frac{6}{u}
   \]

4. **Substitute Back \( u = 6 + \tan^3 x \)**: 
   Substituting \( u \) back in terms of \( x \), we have:
   \[
   -\frac{6}{6 + \tan^3 x} + C
   \]

Thus, the final answer is:

\[
\boxed{D. \, -\frac{6}{6 + \tan^3 x} + C}
\]

---

Do you want any further details or have any questions? Here are 5 related questions:

1. How does substitution work in integrals involving trigonometric functions?
2. Can you explain the chain rule used in the substitution process?
3. What are some common trigonometric identities useful for integration?
4. How do you know when to use a specific substitution method in an integral?
5. What happens if we differentiate the result to verify the solution?

**Tip**: When dealing with integrals involving complicated compositions, substitution is a very powerful tool for simplifying the integrand.

Evaluate the integral: ∫(18 tan²(x) sec²(x)) / (6 + tan³(x))² dx

This tutorial walks through the step-by-step solution of the integral ∫(18 tan²(x) sec²(x)) / (6 + tan³(x))² dx using substitution. Learn how to simplify the integrand and apply trigonometric identities to reach the final solution. Suitable for university-level calculus students.

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