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STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
The integral $\int\limits_{7\pi /4}^{7\pi /3} {\sqrt {{{\tan }^2}\,x}\,dx } $ is equal to
  • A
    $\log \,\,2\sqrt 2 $
  • B
    $\log \,\,2 $
  • C
    $2\log \,\,2 $
  • ✓
    $\log \,\,\sqrt 2 $
✓

Answer

Correct option: D.
$\log \,\,\sqrt 2 $
d
${\rm{Let }}I = \int\limits_{7\pi /4}^{7\pi /3} {\sqrt {{{\tan }^2}x} dx} $

$\int\limits_{7\pi /4}^{7\pi /3} {\tan xdx}  $

$=  - \left. {\log \cos x} \right|_{7\pi /4}^{7\pi /3}$

$=-\left[\log \cos \frac{7 \pi}{3}-\log \cos \frac{7 \pi}{4}\right]$

$=\log \cos \frac{7 \pi}{4}-\log \cos \frac{7 \pi}{3}$

$=\log \left[\frac{\cos \frac{7 \pi}{4}}{\cos \frac{7 \pi}{3}}\right]$

$=\log \left[\frac{\cos \left(2 \pi-\frac{\pi}{4}\right)}{\cos \left(2 \pi+\frac{\pi}{3}\right)}\right]$

$=\log \left(\frac{\cos \frac{\pi}{4}}{\cos \frac{\pi}{3}}\right)$

$=\log \left(\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}\right)$

$=\log \left(\frac{2}{\sqrt{2}}\right)=\log \sqrt{2}$

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