To solve the given integral, we have:

$I = \int \left( x^3 \sin(x) + \frac{4}{x^2} e^{2x} \right) \cos(x^2) \, dx$

This integral is quite complex due to the multiple functions involved, including polynomial, trigonometric, and exponential components. The integral can be separated into two parts for simplification:

$I = \int x^3 \sin(x) \cos(x^2) \, dx + \int \frac{4}{x^2} e^{2x} \cos(x^2) \, dx$

1. Solving $\int x^3 \sin(x) \cos(x^2) \, dx$:

This integral does not have a straightforward antiderivative in terms of elementary functions. Techniques such as integration by parts, substitution, or series expansion may be used for an approximate solution or numerical evaluation.

2. Solving $\int \frac{4}{x^2} e^{2x} \cos(x^2) \, dx$:

Similarly, this integral is also not solvable in terms of elementary functions. We might approach this using numerical methods or special functions for an approximate solution.

Alternative Approach

Given the complexity, applying a numerical method or using symbolic computational tools like Mathematica or Wolfram Alpha for approximations is recommended.

Would you like further details on numerical methods for evaluating this integral, or is there a specific part of the problem you'd like me to focus on?

Related Questions:

1. How do numerical integration techniques work for evaluating complex integrals?
2. Can the integral of a product of trigonometric and polynomial functions be simplified using series expansion?
3. What are special functions, and when are they used in evaluating integrals?
4. How does integration by parts apply to more complex functions?
5. What are some examples of integrals that cannot be solved in terms of elementary functions?

Tip: When dealing with integrals that involve complicated combinations of functions, it's often useful to explore numerical solutions or approximations using Taylor series.