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To solve the integral \[ \int \frac{\sqrt{3x^3 + \frac{x + 2}{2}}}{-2(x \cos 2) \sqrt{4 - x^2}} \, dx \] you can consider simplifying the expression first or using substitution methods, depending on the complexity. However, due to the nature of this specific equation, it might be complex to solve directly by standard algebraic means and might require numerical methods or specific techniques depending on the function values and intervals. Here's a general approach: 1. **Trigonometric Substitution**: If the form permits, use trigonometric identities to simplify parts of the function, especially involving square roots. 2. **Numerical Integration**: If a symbolic solution isn't feasible, numerical methods such as Simpson's rule or numerical solvers can be applied to find approximate integrals over a certain interval. 3. **Software Tools**: Use tools like Mathematica, MATLAB, or Python libraries (Sympy, SciPy) to handle complicated integrals. If you have specific values or intervals, it could be easier to analyze further.