Find the value of integral $$\int^{2\pi}_{-\frac{\pi}{2}}\lfloor \cot^{-1}(x)\rfloor dx$$
$\lfloor x \rfloor = x-\{x\}$ and $0\leq \{x\}<1$
for $0\leq x<\cot(1),\lfloor \cot^{-1}(x)\rfloor = 0$ and for $\cot (1)\leq x<2\pi,\lfloor \cot^{-1}(x)\rfloor = 0$
but i want go further for negative interval of $x,$ could some help with me this.