To solve the integral \( \int 4x^2 \tan(x^3) \, dx \), we can use substitution.

### Step-by-Step Solution:

1. **Substitute** \( u = x^3 \).  
   Then, \( du = 3x^2 \, dx \), so \( \frac{du}{3} = x^2 \, dx \).

2. **Rewrite the integral**:
   \[
   \int 4x^2 \tan(x^3) \, dx = \int 4 \tan(u) \cdot \frac{du}{3}
   \]
   This simplifies to:
   \[
   \frac{4}{3} \int \tan(u) \, du
   \]

3. **Integrate** \( \int \tan(u) \, du \):
   The integral of \( \tan(u) \) is \( \ln|\sec(u)| \).

   Therefore:
   \[
   \frac{4}{3} \int \tan(u) \, du = \frac{4}{3} \ln|\sec(u)| + C
   \]

4. **Substitute back** \( u = x^3 \):
   \[
   \frac{4}{3} \ln|\sec(x^3)| + C
   \]

### Final Answer:
\[
\int 4x^2 \tan(x^3) \, dx = \frac{4}{3} \ln|\sec(x^3)| + C
\]

Would you like any more details, or do you have any questions?

Here are 5 related questions to further your understanding:

1. How would you solve \( \int x^2 \sin(x^3) \, dx \)?
2. Can you evaluate \( \int x^2 \cos(x^3) \, dx \) using a similar method?
3. What happens if you change the integral to \( \int 4x^2 \cot(x^3) \, dx \)?
4. How would you approach \( \int x \tan(x^2) \, dx \)?
5. How does the method change for \( \int x \sec^2(x^2) \, dx \)?

**Tip:** When dealing with trigonometric integrals, substitution is often key, especially when the argument of the trigonometric function is more complex (e.g., \( x^3 \)).

Explore the solution to the integral ∫ 4x^2 tan(x^3) dx using substitution method. Learn how to integrate trigonometric functions with complex arguments. Suitable for advanced high school students and college-level learners.

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