The value of ∫−11x2e[x3]dx, where t denotes the greatest integer ≤t, is

∫-11 x2 ex3 dx =∫-10 x2 e-1 dx+∫01 x2 e0 dx =1ex33-10+x3301 =1e0+13+13=13e+13

The value of ∫−11x2e[x3]dx, where t denotes the greatest integer ≤t, is

∫-11 x2 ex3 dx =∫-10 x2 e-1 dx+∫01 x2 e0 dx =1ex33-10+x3301 =1e0+13+13=13e+13

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The value of $\int_{- 1}^{1} x^{2} e^{[x^{3}]} d x,$ where $[t]$ denotes the greatest integer $\leq t,$ is

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$\frac{e + 1}{3}$

$\frac{e + 1}{3 e}$

$\frac{e - 1}{3 e}$

$\frac{1}{3 e}$

answer is B.

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Detailed Solution

$\begin{array}{rcl} \int_{- 1}^{1} x^{2} e^{[x^{3}]} dx \\ = & \int_{- 1}^{0} x^{2} e^{- 1} dx + \int_{0}^{1} x^{2} e^{0} dx \\ = & \frac{1}{e} {(\frac{x^{3}}{3})}_{- 1}^{0} + {(\frac{x^{3}}{3})}_{0}^{1} \\ = & \frac{1}{e} (0 + \frac{1}{3}) + \frac{1}{3} = \frac{1}{3 e} + \frac{1}{3} \end{array}$