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Answer
$$ \displaystyle\int \sqrt{4-{x}^{2}}{\cdot}\left({x}^{3}{\cdot}\cos\left(\dfrac{x}{2}\right)+\dfrac{1}{2}\right)\, \mathrm d x = {{x\,\sqrt{4-x^2}+4\,\arcsin \left({{x}\over{2}}\right)}\over{4}} $$
Explanation
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