the value of the constant a>0 such that integation from 0 to a[tan-1x^1/2]dx=integation from 0 to a [cot-1x^1/2]dx,where [.] denotes the greatest integer function.


2 years ago

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                    $\hspace{-16}\mathbf{\int_{0}^{a}\left[\tan^{-1}\sqrt{x}\right]dx=\int_{0}^{a}\left[\cot^{-1}\sqrt{x}\right]dx}\\\\\\ Now Here we have to find the value of \mathbf{x} for which\\\\\\ \mathbf{\left[\tan^{-1}\sqrt{x}\right]} and \mathbf{\left[\cot^{-1}\sqrt{x}\right]} is an Integer.So\\\\\\ \mathbf{\int_{0}^{\tan^2 (1)}\left[\tan^{-1}\sqrt{x}\right]dx+\int_{\tan^2 (1)}^{a}\left[\tan^{-1}\sqrt{x}\right]dx}\\\\\\ \mathbf{=\int_{0}^{\cot^2(1)}\left[\cot^{-1}\sqrt{x}\right]dx+\int_{\cot^2(1)}^{a}\left[\cot^{-1}\sqrt{x}\right]dx}\\\\\\ So \mathbf{\int_{0}^{\tan^2(1)}0.dx+\int_{\tan^2(1)}^{a}1.dx=\int_{0}^{\cot^2(1)}1.dx+\int_{\cot^2(1)}^{a}0.dx}\\\\\\ So \mathbf{a-\tan^2(1)=\cot^2(1)}\\\\\\ So \mathbf{a=\tan^2(1)+\cot^2(1)}$

2 years ago

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