First slide

Evaluation of definite integrals

Question

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

Moderate

A

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

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

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

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

Solution

$\begin{matrix} \int_{- 1}^{1} x^{2} e^{[x^{3}]} d x = \int_{- 1}^{0} x^{2} e^{- 1} d x + \int_{0}^{1} x^{2} d x ∣ \\ = \frac{1}{e} {[\frac{x^{3}}{3}]}_{- 1}^{0} + {[\frac{x^{3}}{3}]}_{0}^{1} \\ = \frac{1}{e} (0 - (- \frac{1}{3})) + \frac{1}{3} \\ = \frac{1}{3 e} + \frac{1}{3} \end{matrix}$

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