Question

$\underset{-2}{\overset{1}{\int }}\frac{\left|x\right|}{x}dx.$

Answer 1

$$\begin{gathered} \mathrm{Let},I={\int }_{-2}^1\frac{\left|x\right|}{x}dx \\ \mathrm{We}\mathrm{have}, \\ \left|x\right|=\left\{\begin{array}{l}x0\le x\le 1\\ -x-2\le x<0\end{array}\right. \\ \therefore \frac{\left|x\right|}{x}=\left\{\begin{array}{l}10\le x\le 1\\ -1-2\le x<0\end{array}\right. \\ \mathrm{Therefore}, \\ I={\int }_{-2}^0-1dx+{\int }_0^{1}1dx \\ =-{\left[x\right]}_{-2}^0+{\left[x\right]}_0^1 \\ =0-2+1-0 \\ =-1 \end{gathered}$$