Integral Calculus: JEE Main 2022 Mathematics question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined as $f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}$ where $[t]$ is the greatest integer less than or equal to $t$. If $\mathop {\lim }\limits_{x \to -1 } f(x)$ exists, then the value of $\int\limits_{0}^{4} f(x) d x$ is equal to