<p>The value of \(\displaystyle\int_{-5}^{5} |x + 2|\,dx\) is ______.</p>
Step-by-Step Solution
Key Concept: Split the absolute value at its critical point x = -2 where the expression inside changes sign, then integrate separately on each interval where the expression has constant sign.
<p><strong>Step 1:</strong> Identify where x + 2 = 0, which gives x = -2. This is our critical point where the expression inside the absolute value changes sign.</p><p><strong>Step 2:</strong> For x ∈ [-5, -2]: x + 2 ≤ 0, so |x + 2| = -(x + 2) = -x - 2</p><p>For x ∈ [-2, 5]: x + 2 ≥ 0, so |x + 2| = x + 2</p><p><strong>Step 3:</strong> Split the integral:</p><p>∫₋₅⁵ |x + 2| dx = ∫₋₅⁻² (-x - 2) dx + ∫₋₂⁵ (x + 2) dx</p><p><strong>Step 4:</strong> Evaluate the first integral:</p><p>∫₋₅⁻² (-x - 2) dx = [-x²/2 - 2x]₋₅⁻² = [(-4/2 - 2(-2)) - (-25/2 - 2(-5))]</p><p>= [(-2 + 4) - (-25/2 + 10)] = [2 - (-5/2)] = 2 + 5/2 = 9/2</p><p><strong>Step 5:</strong> Evaluate the second integral:</p><p>∫₋₂⁵ (x + 2) dx = [x²/2 + 2x]₋₂⁵ = [(25/2 + 10) - (4/2 + 2(-2))]</p><p>= [(25/2 + 10) - (2 - 4)] = [45/2 - (-2)] = 45/2 + 2 = 49/2</p><p><strong>Step 6:</strong> Add both parts:</p><p>9/2 + 49/2 = 58/2 = 29</p><p>∴ Answer: <strong>29</strong></p>
Correct Answer: 29