<p>Evaluate \(\displaystyle\int_{-2}^2|x+1|\,dx\)</p>
Step-by-Step Solution
Key Concept: x+1 = 0 at x = -1. Split: \int₋_2^(-1) -(x+1)dx + \int₋_1^2 (x+1)dx.
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<p>Critical point: $x+1=0\Rightarrow x=-1\in[-2,2]$.</p>
<p>$$I = \int_{-2}^{-1}-(x+1)\,dx + \int_{-1}^2(x+1)\,dx$$</p>
<p>$$= \left[-\frac{(x+1)^2}{2}\right]_{-2}^{-1} + \left[\frac{(x+1)^2}{2}\right]_{-1}^2$$</p>
<p>$$= \left(0-\left(-\frac{1}{2}\right)\right) + \left(\frac{9}{2}-0\right) = \frac{1}{2}+\frac{9}{2} = \boxed{5}$$</p>
Correct Answer: D