Question

Evaluate the following integrals:

Answer 1

$$\begin{gathered} {\int }_{-2}^2\left|x+1\right|\mathit{d}x \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|x+1\right|=\left\{\begin{array}{l}-\left(x+1\right),-2\le x\le -1\\ x+1,-1<x\le 2\end{array}\right. \\ \therefore I={\int }_{-2}^2\left|x+1\right|\mathit{d}x \\ \Rightarrow I={\int }_{-2}^{-1}-\left(x+1\right)dx+{\int }_{-1}^2\left(x+1\right)dx \\ \Rightarrow I={\left[\frac{-x^2}{2}-x\right]}_{-2}^{-1}+{\left[\frac{x^2}{2}+x\right]}_{-1}^2 \\ \Rightarrow I=\frac{-1}{2}+1+2-2+2+2-\frac{1}{2}+1 \\ \Rightarrow I=5 \end{gathered}$$