$ \int\limits_1^2 x ^2 e ^{\left[ x ^3\right]+1} dx$
$ x ^3= t$
$ 3 x ^2 dx = dt $
$ =\frac{ e }{3} \int\limits_1^8 e ^{[ t ]} dt $
$=\frac{ e }{3}\left\{\int\limits_1^2 e dt +\int_2^3 e ^2 dt +\ldots \ldots . .+\int\limits_7^8 e ^7 dt \right\} $
$ =\frac{ e }{3}\left( e + e ^2+\ldots \ldots \ldots+ e ^7\right)$
$ =\frac{ e ^2}{3}\left(1+ e +\ldots \ldots \ldots+ e ^6\right)=\frac{ e ^2}{3} \frac{\left( e ^7-1\right)}{( e -1)}$
$ \frac{3( e -1)}{ e } \int\limits_1^2 x ^2 \times e ^{[ x ]+\left[ x ^3\right]} dx =\frac{3}{ e }( e -1) \times \frac{ e ^2}{3} \frac{\left( e ^7-1\right)}{( e -1)}$
$ = e \left( e ^7-1\right) $
$ = e ^8- e $