Question

\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]

Options

• \[\frac{17}{3}\]

• \[\frac{19}{3}\]

• 19

• 17

Answer 1

\[\frac{19}{3}\]

Explanation:

I = \[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]

\[=\int_{-2}^{-1}\left(x^2-x-2\right)\mathrm{d}x+\int_{-1}^2-\left(x^2-x-2\right)\mathrm{d}x\]

\[=\int_{-2}^{-1}\left(x^2-x-2\right)\mathrm{d}x+\int_{-1}^{2}\left(-x^2+x+2\right)\mathrm{d}x\]

\[=\left[\frac{x^3}{3}-\frac{x^2}{2}-2x\right]_{-2}^{-1}+\left[\frac{-x^3}{3}+\frac{x^2}{2}+2x\right]_{-1}^2\]

\[=\frac{-1}{3}-\frac{1}{2}+2-\left(\frac{-8}{3}-2+4\right)-\frac{8}{3}+\frac{4}{2}+4-\left(\frac{1}{3}+\frac{1}{2}-2\right)\]

= \[\frac{19}{3}\]