\( \int_{0}^{2 n \pi}\left(|\sin x|-\left[\left|\frac{\sin x}{2}\right|\right]\right) d x \quad[...
\( \int_{0}^{2 n \pi}\left(|\sin x|-\left[\left|\frac{\sin x}{2}\right|\right]\right) d x \quad[\cdot] \) denotes greatest integer function. Note that: \( \left[\left|\frac{\sin x}{2}\right|\right]=0, n \in N \)
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