Question
The value of $$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin \,x} \right]dx} $$ where [.] represents the greatest integer function is-
A.
$$\frac{{ - 5\pi }}{3}$$
B.
$$ - \pi $$
C.
$$\frac{{ 5\pi }}{3}$$
D.
$$ - 2\pi $$
Answer :
$$\frac{{ - 5\pi }}{3}$$
Solution :
$$\eqalign{
& {\text{Let}} \cr
& I = \int_\pi ^{2\pi } {\left[ {2\,\sin \,x} \right]dx} \cr
& \pi \leqslant x < \frac{{7\pi }}{6}\, \Rightarrow - 1 \leqslant 2\,\sin \,x < 0 \cr
& \Rightarrow \left[ {2\,\sin \,x} \right] = - 1 \cr
& \frac{{7\pi }}{6} \leqslant x < \frac{{11\pi }}{6}\, \Rightarrow - 2 \leqslant 2\,\sin \,x < - 1 \cr
& \Rightarrow \left[ {2\,\sin \,x} \right] = - 1 \cr
& \therefore \,I = \int\limits_\pi ^{\frac{{7\pi }}{6}} { - 1\,dx + } \int\limits_{\frac{{7\pi }}{6}}^{\frac{{11\pi }}{6}} { - 2\,dx + } \int\limits_{\frac{{11\pi }}{6}}^{2\pi } { - 1\,dx} \cr
& = \left( { - \frac{{7\pi }}{6} + \pi } \right) + 2\left( { - \frac{{11\pi }}{6} + \frac{{7\pi }}{6}} \right) + \left( { - 2\pi + \frac{{11\pi }}{6}} \right) \cr
& = - \frac{\pi }{6} - \frac{{8\pi }}{6} - \frac{\pi }{6} \cr
& = - \frac{{10\pi }}{6} \cr
& = \frac{{ - 5\pi }}{3} \cr} $$