\[ f(x)=\sqrt{\dfrac{\left(4-\sqrt{x^2} \right) \left(\sqrt{x^2}+2 \right)}{ \left(10-\sqrt{x^2} \right) \left(\displaystyle \int_{0}^{\frac{\pi}{2}} \sin^9 \theta \ d\theta -\sqrt{x^2} \right)}} \]

How many integral values of \(x\) are included in the domain of the above real-valued function \(f(x)\)?

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