_0^ 6 ^2 (x- 6 ) d x is equal to : - Vidyadip

MCQ

$\int_0^{\frac{\pi}{6}} \sec ^2\left(x-\frac{\pi}{6}\right) d x$ is equal to :

✓
$\frac{1}{\sqrt{3}}$
B
$-\frac{1}{\sqrt{3}}$
C
$\sqrt{3}$
D
$-\sqrt{3}$

✓

Answer

Correct option: A.

$\frac{1}{\sqrt{3}}$

$\text {We have, } \int_0^{\pi / 6} \sec ^2\left(x-\frac{\pi}{6}\right) d x$
$=\left[\tan \left(x-\frac{\pi}{6}\right)\right]_0^{\pi / 6}$
$=\tan \left(\frac{\pi}{6}-\frac{\pi}{6}\right)-\tan \left(0-\frac{\pi}{6}\right)$
$=\tan 0^{\circ}-\tan \left(-\frac{\pi}{6}\right) $
$\left(\because \tan 0^{\circ}=0 \text { and } \tan (-\theta)=-\tan \theta\right)$
$=0+\tan (\pi / 6)=\tan \frac{\pi}{6}$
$=\frac{1}{\sqrt{3}}$

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