Let f:RR be a function defined as f(x) = a[x]2+[2-x], aR, where [t] is the greatest intege…

Question

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

Accepted Answer

For the limit to exist at $x=-1$: $\displaystyle\lim_{x\to-1^-}f(x)=\displaystyle\lim_{x\to-1^+}f(x)$, i.e. $a\sin(-\pi)+3 = a\sin\left(-\dfrac\pi2\right)+2$, giving $3=-a+2 \implies a=-1$.

$$\int_0^4 f(x)\,dx = \int_0^4\left(-\sin\frac{\pi[x]}2+[2-x]\right)dx = \int_0^1 1\,dx+\int_1^2(-1)\,dx+\int_2^3(-1)\,dx+\int_3^4(-1)\,dx$$

$= 1-1-1-1 = -2$

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