Answer
$$ \displaystyle\int {x}^{2}{\cdot}{\left(\cos\left(x\right)\right)}^{2}\, \mathrm d x = {{6\,x^2\,\sin \left(2\,x\right)-3\,\sin \left(2\,x\right)+6\,x\, \cos \left(2\,x\right)+4\,x^3}\over{24}} $$
Explanation
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