Let $\mathrm{f} : \mathrm{ℝ}\to \mathrm{ℝ}$ be a function defined as $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{asin}\left(\frac{\mathrm{\pi }\left[\mathrm{x}\right]}{2}\right)+[2-\mathrm{x}], \mathrm{a}\in \mathrm{ℝ}$, where [t] is the greatest integer less than or equal to t. If $\underset{\mathrm{x}\to -1}{lim} \mathrm{f}\left(\mathrm{x}\right)$ exists, then the value of ${\int }_0^4 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

$\underset{\mathrm{x}\to -1^{+}}{lim} \mathrm{asin}\left(\mathrm{\pi }\frac{\left[\mathrm{x}\right]}{2}\right)+[2-\mathrm{x}]=-\mathrm{a}+2$

$\underset{\mathrm{x}\to -1^{-}}{lim} \mathrm{asin}\left(\mathrm{\pi }\frac{\left[\mathrm{x}\right]}{2}\right)+[2-\mathrm{x}]=0+3=3$

$$\begin{gathered} \underset{\mathrm{x}\to -1}{lim} \mathrm{f}\left(\mathrm{x}\right)\text{ exist when } \\ -a+2=3 \\ \Rightarrow \mathrm{a}=-1 \end{gathered}$$

Now, 

$\begin{array}{l}{\int }_0^4 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^1 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_1^2 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_2^3 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_3^4 \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\\ ={\int }_0^1 (0+1)\mathrm{dx}+{\int }_1^2 (-1+0)\mathrm{dx}+{\int }_2^3 (0-1)\mathrm{dx}+{\int }_3^4 (1-2)\mathrm{dx}\\ =1-1-1-1=-2\end{array}$