Evaluate $\int\limits_{-1}^{2} \frac{|x|}{x} dx$.

1

The correct answer is Option (2) → 1

Let $I = \int\limits_{-1}^{2} \frac{|x|}{x} dx$.

Since $\frac{|x|}{x} = \begin{cases} \frac{-x}{x}, & x < 0 \\ \frac{x}{x}, & x > 0 \end{cases}$

$= \begin{cases} -1, & x < 0 \\ 1, & x > 0 \end{cases}$

$∴I = \int\limits_{-1}^{0} (-1) dx + \int_{0}^{2} (1) dx$

$I = [-x]_{-1}^{0} + [x]_{0}^{2}$

$I = [0 - (-(-1))] + [2 - 0]$

$I = -1 + 2 = 1$