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  4. If [.] denotes the greatest integer function then ∫ limitsπ2 π[2 sin x] d x is equal to

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Q. If [.] denotes the greatest integer function then $\int\limits_\pi^{2 \pi}[2 \sin x] d x$ is equal to

Integrals

Solution:

$I=\int\limits_\pi^{\pi+\pi / 6}(-1) d x+\int\limits_{\pi+\frac{\pi}{6}}^{\pi+\frac{5 \pi}{6}}(-2) d x+\int\limits_{\pi+\frac{5 \pi}{6}}^{2 \pi}(-1) d x$
$=-\frac{\pi}{6}-2\left(\frac{4 \pi}{6}\right)-\left(\frac{\pi}{6}\right)=-\frac{5 \pi}{3}$