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The value of \( \int_{0}^{2 \pi}\left[\cot ^{-1} x\right] d x \) where [.] denotes greatest integer function, is.... |
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Answer» \(\int^{2\pi}_0[\cot^{-1}\,x]dx\) \(= \int^{\cot 1}_0[\cot^{-1}\,x]dx\) \(+\, \int^{2\pi}_{\cot 1}[\cot^{-1}\,x]dx\) \(= \int^{\cot 1}_0 1\,\text{dx}\) \(+\, \int^{2\pi}_{\cot 1} 0\,\text{dx}\) (\(\because\) when \(x \in [0, \cot 1],\) \(\cot^{-1}x \in \left[\frac{\pi}{2}, 1\right]\) \(\Rightarrow [\cot^{-1} x] = 1\) and when \(x \in [\cot 1, 2\pi],\) \(\cot^{-1}x \in [1, 0]\) \(\Rightarrow [\cot^{-1} x] = 0\)) \(= [x]^{\cot 1}_0 = \cot 1 - 0 = \cot 1.\) |
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