First slide

Evaluation of definite integrals

Question

$Evaluate \int_{- 1}^{2} |x^{3} - x| d x$

Moderate

A

$\frac{11}{4}$
B

$\frac{11}{3}$
C

$\frac{3}{11}$
D

$\frac{2}{11}$

Solution

$f (x) = x^{3} - x = x (x^{2} - 1) = x (x - 1) (x + 1) Sign scheme of f (x) is as shown in the following figure.$
$From the sign scheme, we have$
$\begin{matrix} \int_{- 1}^{2} |x^{3} - x| d x = \int_{- 1}^{0} (x^{3} - x) d x + \int_{0}^{1} - (x^{3} - x) d x + \int_{1}^{2} (x^{3} - x) d x \\ = {[\frac{x^{4}}{4} - \frac{x^{2}}{2}]}_{- 1}^{0} + {[\frac{x^{2}}{2} - \frac{x^{4}}{4}]}_{0}^{1} + {[\frac{x^{4}}{4} - \frac{x^{2}}{2}]}_{1}^{2} \\ = - (\frac{1}{4} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{4}) + (4 - 2) - (\frac{1}{4} - \frac{1}{2}) \\ = - \frac{1}{4} + \frac{1}{2} + \frac{1}{2} - \frac{1}{4} + 2 - \frac{1}{4} + \frac{1}{2} = \frac{11}{4} \end{matrix}$

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