question_answer

Evaluate \[\int_{-2}^{1}{\frac{|x|}{x}\,}dx.\]

Answer:

Let \[l=\int_{-2}^{1}{\frac{|x|}{x}dx=\int_{-2}^{0}{\frac{|x|}{x}dx}+\int_{0}^{1}{\frac{|x|}{x}dx}}\]   \[=\int_{-2}^{0}{\frac{-x}{x}\,}dx+\int_{0}^{1}{\frac{x}{x}\,}dx\]                \[\left[ \because \,\,\,|x|=\left\{ \begin{matrix}    -x,x<0  \\    x,x\ge 0  \\ \end{matrix} \right. \right]\] \[=\int_{-2}^{0}{-1dx+\int_{0}^{1}{1dx=[-x]_{-2}^{0}+[x]_{0}^{1}}}\] \[=[0-(+2)]+[1-0]=-\,2+1\] \[=-1\]