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The problem is asking to find an asymptotically equivalent function of the form \( ax^\alpha \) as \( x \to +\infty \) for the given function:
\[
f(x) = \ln x^5 + 2x^3 + 2 \sin^6 (x \cdot \arctan x)
\]

### Step-by-step breakdown:

1. **First term: \( \ln x^5 \)**
   - \( \ln x^5 = 5 \ln x \).
   - As \( x \to +\infty \), \( \ln x \) grows very slowly compared to powers of \( x \), so this term is negligible in comparison to the others.

2. **Second term: \( 2x^3 \)**
   - This is a polynomial term, and as \( x \to +\infty \), this term grows as \( 2x^3 \).

3. **Third term: \( 2 \sin^6 (x \cdot \arctan x) \)**
   - For large \( x \), \( \arctan x \approx \frac{\pi}{2} \), so \( x \cdot \arctan x \approx \frac{\pi x}{2} \).
   - The sine of large multiples of \( x \) oscillates between -1 and 1, so \( \sin^6(x \cdot \arctan x) \) oscillates between 0 and 1.
   - Therefore, \( 2 \sin^6(x \cdot \arctan x) \) is bounded and oscillatory, and it does not contribute to the asymptotic growth compared to \( x^3 \).

### Conclusion:
The dominant term as \( x \to +\infty \) is \( 2x^3 \). Therefore, the function \( f(x) \) is asymptotically equivalent to \( 2x^3 \), which corresponds to option **c**.

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Here are 5 related questions you might want to explore:
1. How does \( \arctan(x) \) behave as \( x \to \infty \)?
2. Why is the logarithmic term \( \ln x^5 \) considered negligible compared to \( x^3 \)?
3. How would the result change if the sine term had a different power (e.g., \( \sin^2(x \cdot \arctan x) \))?
4. What is the meaning of "asymptotically equivalent" when comparing functions?
5. How does one deal with oscillatory terms in asymptotic analysis?

**Tip:** Always focus on the term with the highest growth rate when analyzing the behavior of functions as \( x \to +\infty \).

Tìm một hàm số tương đương dạng ax^α khi x → +∞ của f(x) = ln x^5 + 2x^3 + 2 sin^6(x arctan x)

Find the asymptotically equivalent function for f(x) = ln(x^5) + 2x^3 + 2sin^6(x arctan x) as x → +∞. Explore how to ignore lower-growth and oscillatory terms, focusing on dominant terms in asymptotic analysis.

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