The value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\displaystyle\mathrm{cosec}^{2}x\,dx\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\sqrt{3}\)
Since \(\frac{d}{dx}(\cot x) = -\mathrm{cosec}^{2}x\), an antiderivative of \(\mathrm{cosec}^{2}x\) is \(-\cot x\). So
\( \displaystyle \int_{\pi/6}^{\pi/2}\mathrm{cosec}^{2}x\,dx = \left[-\cot x\right]_{\pi/6}^{\pi/2} \)
\( \displaystyle = -\cot\frac{\pi}{2} + \cot\frac{\pi}{6} \)
\( = -0 + \sqrt{3} = \sqrt{3} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{\sqrt{3}}\) — Incorrect. This is \(\tan\frac{\pi}{6}\); the value needed is \(\cot\frac{\pi}{6} = \sqrt{3}\).
Option (2) \(-\sqrt{3}\) — Incorrect. It uses \(+\cot x\) as the antiderivative; the integrand \(\mathrm{cosec}^{2}x\) is positive, so the value must be positive.
Option (4) 2 — Incorrect. This is \(\mathrm{cosec}\frac{\pi}{6}\), not the value of \(-\cot x\) between the limits.