Question

Evaluate the following integrals:
$\underset{-6}{\overset{6}{\int }}\left|x+2\right|dx$

Answer 1

$$\begin{gathered} {\int }_{-6}^6\left|x+2\right|dx \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|x+2\right|=\left\{\begin{array}{l}-\left(x+2\right),-6\le x\le -2\\ x+2,-2<x\le 6\end{array}\right. \\ \therefore I={\int }_{-6}^6\left|x+2\right|\mathit{d}x \\ \Rightarrow I={\int }_{-6}^{-2}-\left(x+2\right)dx+{\int }_{-2}^6\left(x+2\right)dx \\ \Rightarrow I={\left[\frac{-x^2}{2}-2x\right]}_{-6}^{-2}+{\left[\frac{x^2}{2}+2x\right]}_{-2}^6 \\ \Rightarrow I=-2+4+18-12+18+12-2+4 \\ \Rightarrow I=40 \end{gathered}$$