\(\int_{\pi / 3}^{\pi / 2} x \sin (\pi[x]-x) d x \text { is equal to : }\)

\(\frac{1}{2}+\frac{\pi}{6}\)

\(1-\frac{\sqrt{3}}{2}+\frac{\pi}{6}\)

\(-\frac{1}{2}-\frac{\pi}{6}\)

\(\frac{\sqrt{3}}{2}-1-\frac{\pi}{6}\)