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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
$\int_0^{\frac{\pi}{6}} \sec ^2\left(x-\frac{\pi}{6}\right) d x$ is equal to :
  • A
    $\frac{1}{\sqrt{3}}$
  • B
    $-\frac{1}{\sqrt{3}}$
  • C
    $\sqrt{3}$
  • D
    $-\sqrt{3}$
✓

Answer

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

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