MCQ
$\int\limits_{7\pi /4}^{7\pi /3} {\sqrt {{{\tan }^2}\,x}\,dx } $ મેળવો.
- A$\log \,\,2\sqrt 2 $
- B$\log \,\,2 $
- C$2\log \,\,2 $
- ✓$\log \,\,\sqrt 2 $
$\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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