To solve the integral $\int \left( x^3 \cos \left( \frac{x}{2} \right) + \frac{1}{2} \right) \sqrt{4 - x^2} \, dx$, we will proceed step by step.

Step 1: Split the Integral

First, break the integral into two parts:

$I = \int \left( x^3 \cos \left( \frac{x}{2} \right) + \frac{1}{2} \right) \sqrt{4 - x^2} \, dx$

This can be written as:

$I = \int x^3 \cos \left( \frac{x}{2} \right) \sqrt{4 - x^2} \, dx + \int \frac{1}{2} \sqrt{4 - x^2} \, dx$

We now handle each term separately.

Step 2: Solve the Second Integral

Start with the simpler term:

$\int \frac{1}{2} \sqrt{4 - x^2} \, dx$

This is a standard integral, and it can be solved by a trigonometric substitution:

Let $x = 2 \sin \theta$, so that $dx = 2 \cos \theta \, d\theta$, and $\sqrt{4 - x^2} = 2 \cos \theta$. The integral becomes:

$\frac{1}{2} \int 2 \cos^2 \theta \cdot 2 \cos \theta \, d\theta = 2 \int \cos^2 \theta \, d\theta$

Now use the identity $\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$:

$2 \int \cos^2 \theta \, d\theta = \int (1 + \cos(2\theta)) \, d\theta$

This integrates to:

$\theta + \frac{1}{2} \sin(2\theta) + C$

Converting back to $x$, recall that $\theta = \arcsin \left( \frac{x}{2} \right)$, and we get:

$\frac{x}{2} \sqrt{4 - x^2} + C_1$

Step 3: First Integral

Now handle the more complicated term:

$\int x^3 \cos \left( \frac{x}{2} \right) \sqrt{4 - x^2} \, dx$

This integral is more challenging and may require advanced techniques such as integration by parts or a series expansion. For now, this can be computed using numerical methods or specialized software for exact solutions, as it does not have a simple closed form.

Step 4: Combine the Results

Thus, the solution to the integral is:

$I = \frac{x}{2} \sqrt{4 - x^2} + \text{(solution to the first integral)} + C$

Let me know if you'd like help finding the numerical value of the first term or clarifying any step.

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Would you like more details or have any questions?
Here are some related problems to explore:

1. Integrate $\int \sqrt{4 - x^2} \, dx$.
2. Solve $\int x^2 \cos(x) \, dx$.
3. Compute $\int \sin(x) \sqrt{1 - x^2} \, dx$.
4. Find the integral $\int x^3 e^x \, dx$.
5. Evaluate $\int x^3 \sqrt{4 - x^2} \, dx$.

Tip: Breaking a complex integral into smaller parts is often key to finding a solution.