https://file.aiquickdraw.com/mathbot/file/1725632815649covxi65z.jpg

The integral provided in the image is:

\[
\int \sin(2x) \cdot \cos(x^3) \, dx
\]

This integral involves a product of trigonometric and non-elementary functions, making it difficult to solve with standard methods. Here's how we can approach this:

### Substitution attempt:
Let’s consider possible substitutions to simplify the integral. However, there is no straightforward substitution that makes this integrable using elementary functions.

This type of integral may require numerical methods or special functions (e.g., Fresnel or hypergeometric functions) for an explicit solution. As it stands, the integral cannot be simplified into elementary functions.

Would you like me to compute an approximate numerical solution for this integral or explore further theoretical details?

### Related Questions:
1. How do we evaluate non-elementary integrals in general?
2. What are special functions, and how are they used in integration?
3. How does substitution help simplify integrals?
4. When is numerical integration more effective than symbolic integration?
5. What methods exist for approximating complex integrals?

**Tip:** When encountering integrals that are difficult or impossible to solve with elementary functions, look into numerical approaches like Simpson's rule or Gaussian quadrature!

Explore the integral of sin(2x) * cos(x^3) and how it involves non-elementary functions. Learn about special functions, numerical integration techniques, and possible approaches for solving complex integrals. Ideal for university-level calculus students.

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