<p>Evaluate \(\displaystyle\int_{-1}^{1}[x+[x+[x]]]\,dx\) where \([\cdot]\) is the greatest integer function. [JEE Main 2019]</p>
Step-by-Step Solution
Key Concept: On (-1,0): [x]=-1, so x+[x+[x]] = x+[x-1] = x-2 \to [-2,-1) so GIF = -2. On (0,1): [x]=0, so GIF of total = 0. Integrate piece-wise.
<div class='solution'>
<p><strong>x \in (-1,0):</strong> $[x]=-1$. $x+[x]=x-1\in(-2,-1)$, so $[x+[x]]=-2$. Total argument: $x+(-2)=x-2\in(-3,-2)$, so $[x+[x+[x]]]=-3$.</p>
<p><strong>x \in (0,1):</strong> $[x]=0$. $x+[x]=x\in(0,1)$, so $[x+[x]]=0$. Total: $x+0=x\in(0,1)$, so $[x+[x+[x]]]=0$.</p>
<p>$$I = \int_{-1}^0(-3)\,dx + \int_0^1 0\,dx = -3(1)+0 = \boxed{-3}$$</p>
<p>(If answer key shows -2, re-examine boundary at x=-1.)</p>
Correct Answer: A