Value of \( \int_{-2}^{2}\displaystyle|2x+1|dx \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{17}{2} \)
2x + 1 changes sign at \( x=-\frac{1}{2} \):
\( \displaystyle |2x+1|=-(2x+1) \text{ for } x\lt -\frac{1}{2} \)
\( \displaystyle |2x+1|=2x+1 \text{ for } x\ge -\frac{1}{2} \)
\( \displaystyle \int_{-2}^{2}\displaystyle|2x+1|\,dx=\int_{-2}^{-1/2}\displaystyle -(2x+1)\,dx+\int_{-1/2}^{2}\displaystyle(2x+1)\,dx \)
\( \displaystyle =-\Big[x^2+x\Big]_{-2}^{-1/2}+\Big[x^2+x\Big]_{-1/2}^{2} \)
At \( x=-\frac{1}{2} \): \( x^2+x=\frac{1}{4}-\frac{1}{2}=-\frac{1}{4} \); at x = −2: 2; at x = 2: 6.
\( \displaystyle =-\left(-\frac{1}{4}-2\right)+\left(6+\frac{1}{4}\right) \)
\( \displaystyle =\frac{9}{4}+\frac{25}{4} \)
\( \displaystyle =\frac{34}{4} \)
\( \displaystyle =\frac{17}{2} \)
Shortcut: The graph of |2x + 1| is two triangles with vertex at \( x=-\frac{1}{2} \): on the left, base \( \frac{3}{2} \) and height |−4 + 1| = 3, area \( \frac{9}{4} \); on the right, base \( \frac{5}{2} \) and height 5, area \( \frac{25}{4} \). Total \( \frac{17}{2} \).