Question

Evaluate the following integrals:
$\underset{-3}{\overset{3}{\int }}\left|x+1\right|dx$

Answer 1

$$\begin{gathered} I={\int }_{-3}^3\left|x+1\right|\mathit{d}x \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|x+1\right|=\left\{\begin{array}{l}-\left(x+1\right),-3\le x\le -1\\ x+1,-1<x\le 3\end{array}\right. \\ \therefore I={\int }_{-3}^{-1}-\left(x+1\right)\mathit{}\mathit{d}x+{\int }_{-1}^3\left[x+1\right]\mathit{d}x \\ \Rightarrow I={\left[-\frac{{\left(x+1\right)}^2}{2}\right]}_{-3}^{-1}+{\left[\frac{{\left(x+1\right)}^2}{2}\right]}_{-1}^3 \\ \Rightarrow I=0+2+8-0 \\ \Rightarrow I=10 \end{gathered}$$