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The integral you provided is:

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

### Steps to solve:

1. **Symmetry Consideration**:
   The limits of integration are symmetric about zero (from \(-2\) to \(2\)), and the integrand involves both even and odd functions:
   - \(x^3 \cos\left( \frac{x}{2} \right)\) is an odd function (since \(x^3\) is odd, and cosine is an even function).
   - \(\frac{1}{2} \sqrt{4 - x^2}\) is an even function.
   
   Using this, we can simplify the integral:
   - The integral of an odd function over a symmetric interval \([-a, a]\) is zero.
   - So, the term involving \(x^3 \cos \left( \frac{x}{2} \right)\) vanishes, and we are left with only the even part of the integrand:
   
   \[
   \int_{-2}^{2} \frac{1}{2} \sqrt{4 - x^2} \, dx
   \]

2. **Evaluate the Remaining Integral**:
   The integral we are now solving is:

   \[
   \frac{1}{2} \int_{-2}^{2} \sqrt{4 - x^2} \, dx
   \]

   This is a standard integral that represents the area of a semicircle with radius 2. The formula for the area of a semicircle of radius \(R\) is:

   \[
   \text{Area} = \frac{1}{2} \pi R^2
   \]

   In this case, \(R = 2\), so the area is:

   \[
   \text{Area} = \frac{1}{2} \pi (2^2) = 2\pi
   \]

3. **Final Result**:
   Therefore, the value of the original integral is:

   \[
   \boxed{2\pi}
   \]

Would you like more details on any part of this solution?

### Related Questions:
1. How do you determine whether a function is odd or even?
2. What are some techniques to solve definite integrals using symmetry?
3. How do you compute the integral of \(\sqrt{4 - x^2}\) geometrically?
4. Can this integral be solved using a trigonometric substitution instead of geometry?
5. What is the role of the cosine function in integrals involving oscillatory behavior?

### Tip:
When solving integrals with symmetric limits, always check for odd and even functions. This can significantly simplify your work!

Evaluate the integral: ∫ from -2 to 2 of (x^3 * cos(x/2) + 1/2) * sqrt(4 - x^2) dx

Learn how to evaluate the definite integral ∫ from -2 to 2 of (x^3 * cos(x/2) + 1/2) * sqrt(4 - x^2) dx by leveraging symmetry. This solution explores the behavior of odd and even functions and uses geometric principles for simplification.

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