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Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals1 Mark
Question
$\int_0^{\frac{\pi}{6}} \sec ^2\left(x-\frac{\pi}{6}\right) d x$ is equal to :
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Answer

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

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