Absolute value of sum of all integral values of ‘$x$’ in the domain of $f\left(x\right)={\cot }^{-1}\left(\sqrt{x\left(x+3\right)}\right)+{\cos }^{-1}\left(\sqrt{x^2+3x+1}\right)$is

$\therefore x\left(x+3\right)\ge 0\Rightarrow x\ge 0$or$x\le -3\mathrm{.......}\left(1\right)$

also $0\le x^2+3x+1\le 1$$\Rightarrow x^2+3x\le 0$

$\Rightarrow x\left(x+3\right)\le 0 and0\le x\le \mathrm{3.......}\left(2\right)$

$\Rightarrow x\left(x+3\right)=0$

$\Rightarrow$ $x=\left\{0,-3\right\}$

$Sum of integral values of x =0-3=-3$

$\therefore$$absolute value of sum of integral values of a =\left|-3\right|=3$