The least integral value of $x$ for which  ${\tan }^{-1}x>{\cot }^{-1}x$ is

$$\begin{gathered} We have {\tan }^{-1}x>co{t}^{-1}x \\ \Rightarrow {\tan }^{-1}x>\frac{\mathrm{\pi }}{2}-{\tan }^{-1}x \\ \Rightarrow 2{\tan }^{-1}x>\frac{\pi }{2} \\ \Rightarrow {\tan }^{-1}x>\frac{\pi }{4} \\ \Rightarrow x>\tan \frac{\pi }{4} \\ \Rightarrow x>1 \\ \therefore the least integral value of x is 2 \end{gathered}$$