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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
The value of $\int_\pi ^{2\pi } {[2\sin x]\,dx,} $ where $[\,\,.\,\,]$ represents the greatest integer function, is
  • A
    $ - \pi $
  • B
    $ - 2\pi $
  • ✓
    $ - \frac{{5\pi }}{3}$
  • D
    $\frac{{5\pi }}{3}$
✓

Answer

Correct option: C.
$ - \frac{{5\pi }}{3}$
c
(c) $\int_\pi ^{2\pi } {[2\sin x]dx = \int_\pi ^{\pi + (\pi /6)} {( - 1)dx + \int_{\pi + (\pi /6)}^{\pi + (\pi /2)} {\,( - 2)dx} } } $

$ + \int_{\pi + (\pi /2)}^{\pi + (\pi /2) + (\pi /3)} {\,( - 2)dx + \int_{\pi + (\pi /2) + (\pi /3)}^{2\pi } {\,( - 1)dx} } $

$ = - \frac{\pi }{6} - 2\left[ {\frac{\pi }{2} - \frac{\pi }{6}} \right] - 2\left[ {\frac{\pi }{3}} \right] - 1\left[ {\frac{\pi }{2} - \frac{\pi }{3}} \right]$

$ = - \frac{\pi }{6} - \frac{{2\pi }}{3} - \frac{{2\pi }}{3} - \frac{\pi }{6}$

$ = - \frac{\pi }{6} - \frac{{8\pi }}{6} - \frac{\pi }{6}$

$= - \frac{{10\pi }}{6} = - \frac{{5\pi }}{3}$.

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