The value of $\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$, where [ t ] denotes the greatest integer $\le$ t, is :

$I = \int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}dx}$ <br><br>Here -1 $\le$ x $\le$ 1 then -1 $\le$ x³ $\le$ 1 <br><br>Integer between -1 and 1 is 0. So integration will be divided into two parts, -1 to 0 and 0 to 1. <br><br>$$ = \int\limits_{ - 1}^0 {{x^2}{e^{[{x^3}]}}dx} + \int\limits_0^1 {{x^2}{e^{[{x^3}]}}dx} $$<br><br>$= \int\limits_{ - 1}^0 {{x^2}{e^{ - 1}}dx} + \int\limits_0^1 {{x^2}{e^0}dx}$<br><br>= $${1 \over e} \times \left[ {{{{x^3}} \over 3}} \right]_{ - 1}^0 + \left[ {{{{x^3}} \over 3}} \right]_0^1$$<br><br>$$ = {1 \over e} \times \left( {0 - \left( {{{ - 1} \over 3}} \right)} \right) + {1 \over 3}$$<br><br>$= {1 \over {3e}} + {1 \over 3} = {{1 + e} \over {3e}}$