Let f: R → R be a function defined as f(x)=a sin ((π[x])/2)+[2-x], a ∈ R, where [t] is the greatest integer less than or equal to t. If lim_x →-1…

Let $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined as $\displaystyle f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}$, where $\displaystyle [t]$ is the greatest integer less than or equal to $\displaystyle t$. If $\displaystyle \lim _{x \rightarrow-1} f(x)$ exists, then the value of $\displaystyle \int_0^4 f(x) d x$ is equal to