Question
Evaluate the following integrals:$\int\limits^1_{-2}\frac{|\text{x}|}{\text{x}}\text{ dx}$
✓
Answer
Let $\int\limits^1_{-2}\frac{|\text{x}|}{\text{x}}\text{ dx}$ We have,$|\text{x}|=\begin{cases}\text{x},&0\leq\text{x}\leq1\\-\text{x},&-2\leq\text{x}<0\end{cases}$
$\therefore\ \frac{|\text{x}|}{\text{x}}=\begin{cases}1,&0\leq\text{x}\leq1\\-1,&-2\leq\text{x}<0\end{cases}$
Therefore,$\text{I}=\int\limits^0_{-2}-1\text{ dx}+\int\limits^1_01\text{ dx}$
$=-\big[\text{x}\big]^0_{-2}+\big[\text{x}\big]_0^1$
$=0-2+1-0$
$=-1$
$\therefore\ \frac{|\text{x}|}{\text{x}}=\begin{cases}1,&0\leq\text{x}\leq1\\-1,&-2\leq\text{x}<0\end{cases}$
Therefore,$\text{I}=\int\limits^0_{-2}-1\text{ dx}+\int\limits^1_01\text{ dx}$
$=-\big[\text{x}\big]^0_{-2}+\big[\text{x}\big]_0^1$
$=0-2+1-0$
$=-1$
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