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The given integral is:

\[
\int_{-2}^{2} \left( x^3 \cos\left(\frac{x}{2}\right) + \frac{1}{2} \right) \sqrt{4 - x^2} \, dx
\]

This is a definite integral with an upper limit of 2 and a lower limit of -2. The integrand involves a combination of trigonometric, polynomial, and square root functions.

Let me analyze this step by step and proceed with evaluating it. 

First, we notice that the integral contains the term \( \sqrt{4 - x^2} \), which resembles the form of a standard trigonometric substitution related to the integral of functions involving the square root of a difference of squares. This suggests that a substitution like \( x = 2 \sin(\theta) \) could be helpful. I'll begin by setting up this substitution and working through the solution.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can attempt to solve this integral by considering trigonometric substitution due to the term \( \sqrt{4 - x^2} \), which often simplifies with the substitution \( x = 2 \sin(\theta) \). Would you like help with that substitution or further steps?

Let me know if you'd like more details, and here are five related questions to expand on this problem:

1. How can trigonometric substitution simplify integrals involving square roots?
2. What is the general form of integrals involving \( \sqrt{a^2 - x^2} \)?
3. How can symmetry in definite integrals help simplify calculations?
4. What techniques can be used for integrals involving both trigonometric and polynomial functions?
5. What is the physical or geometrical interpretation of integrals involving square roots like \( \sqrt{4 - x^2} \)?

**Tip:** When integrating over symmetric limits (like from -2 to 2), check if the integrand is even or odd—it can simplify your work significantly.

∫ from -2 to 2 (x^3 * cos(x/2) + 1/2) * sqrt(4 - x^2) dx

This page provides a detailed solution for the definite integral ∫ from -2 to 2 of (x^3 * cos(x/2) + 1/2) * sqrt(4 - x^2) dx, using trigonometric substitution and symmetry principles. Learn how to simplify and solve this integral, which involves both polynomial and trigonometric functions.

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