If [ t denotes the greatest integer ≤ 1 , then the value of 3(e -...

Let

$I = {∫}_{1}^{2}{x}^{2}{e}^{\left[x\right] + \left[{x}^{3}\right]}dx$

$⇒ I = {∫}_{1}^{2}{x}^{2}{e}^{\left[{x}^{3}\right] + 1}dx$

$⇒ I = e{∫}_{1}^{2}{x}^{2}{e}^{\left[{x}^{3}\right]}dx$

Put ${x}^{3} = t$

$⇒ 3{x}^{2}dx = dt$

So,

$I = \dfrac{e}{3}{∫}_{1}^{8}{e}^{\left[t\right]}dt$

$⇒ I = \dfrac{e}{3}\left\{{∫}_{1}^{2}edt + {∫}_{2}^{3}{e}^{2}dt + \ldots \ldots \ldots + {∫}_{7}^{8}{e}^{7}dt\right\}$

$⇒ I = \dfrac{e}{3}\left(e + {e}^{2} + \ldots \ldots . . + {e}^{7}\right)$

$⇒ I = \dfrac{{e}^{2}}{3}\dfrac{\left({e}^{7} - 1\right)}{( e - 1 )}$

So,

$\dfrac{3 ( e - 1{)}^{2}}{e}{∫}_{1}^{2}{x}^{2}{e}^{\left[x\right] + \left[{x}^{3}\right]}dx = \dfrac{3}{e}\left(e - 1\right) \times \dfrac{{e}^{2}}{3}\dfrac{\left({e}^{7} - 1\right)}{( e - 1 )}$

$⇒ \dfrac{3 ( e - 1{)}^{2}}{e}{∫}_{1}^{2}{x}^{2}{e}^{\left[x\right] + \left[{x}^{3}\right]}dx = {e}^{8} - e$