The value of the definiteintegral \( \int_{0}^{2 n \pi} \max .\left(\sin x, \sin ^{-1}(\sin x)\r...

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The value of the definiteintegral \( \int_{0}^{2 n \pi} \max .\left(\sin x, \sin ^{-1}(\sin x)\right) d x \) equals (where \( n \in I \) )
(a) \( \frac{n\left(\pi^{2}-4\right)}{2} \)
(b) \( \frac{n\left(\pi^{2}-4\right)}{4} \)
(c) \( \frac{n\left(\pi^{2}-8\right)}{4} \)
(d) \( \frac{n\left(\pi^{2}-2\right)}{4} \)
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