The integral _ 7 /4 ^ 7 /3 ^2 ,x ,dx is equal to - Vidyadip

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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