JEE Main 2021 (Online) 25th February Morning ShiftQuestion 1
The value of $\int_{-1}^{1}x^{2}e^{[x^{3}]}dx$, where $[t]$ denotes the greatest integer $\le t$, is :
Definite Integration
Solution
Concept. $[x^3]=-1$ on $[-1,0)$ and $0$ on $[0,1)$; integrate $x^2$ times a constant on each part. Formulas used. (1) $x\in[-1,0)\Rightarrow x^3\in[-1,0)$; $x\in[0,1)\Rightarrow x^3\in[0,1)$Step 1 - Left part. By (1), $e^{-1}\int_{-1}^0x^2dx=\frac1{3e}$. Step 2 - Right part. $e^0\int_0^1x^2dx=\frac13$. Step 3 - Add. $\frac13\Big(1+\frac1e\Big)=\frac{e+1}{3e}$. Answer: (D)