\(\int_{-1}^{2}\displaystyle\frac{|x|}{x}\,dx\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1
Since \(|x| = -x\) for \(x \lt 0\) and \(|x| = x\) for \(x \gt 0\),
\( \displaystyle \frac{|x|}{x} = -1 \text{ for } x \lt 0, \qquad \frac{|x|}{x} = 1 \text{ for } x \gt 0 \)
Splitting the interval at 0:
\( \displaystyle \int_{-1}^{2}\frac{|x|}{x}\,dx = \int_{-1}^{0}(-1)\,dx + \int_{0}^{2}1\,dx \)
\( = -1 + 2\)
\( = 1\)
Why the other options are incorrect
Option (2) −1 — Incorrect. This is only the part from −1 to 0.
Option (3) 3 — Incorrect. It takes the integrand as 1 on the whole interval, ignoring that it is −1 for negative x.
Option (4) 0 — Incorrect. The interval is not symmetric about 0, so the two parts do not cancel.