<p>Find the value of \( \int_{-5}^{5} |x + 2| \, dx \).</p>
Step-by-Step Solution
Key Concept: Split the absolute value at its critical point x = -2 where (x+2) changes sign, then integrate separately on each interval using the definition of absolute value: |x+2| = (x+2) when x ≥ -2 and |x+2| = -(x+2) when x < -2.
<p><strong>Step 1:</strong> Identify the critical point where x + 2 = 0, which is x = -2. This divides the integral into two regions.</p><p><strong>Step 2:</strong> For x ∈ [-5, -2]: x + 2 < 0, so |x+2| = -(x+2) = -x - 2</p><p>∫₋₅⁻² |x+2| dx = ∫₋₅⁻² (-x-2) dx = [-x²/2 - 2x]₋₅⁻² = [(-4/2) - (-10)] - [(-25/2) - (-10)] = (-2 + 10) - (-25/2 + 10) = 8 - (-5/2) = 8 + 5/2 = 21/2</p><p><strong>Step 3:</strong> For x ∈ [-2, 5]: x + 2 ≥ 0, so |x+2| = x + 2</p><p>∫₋₂⁵ |x+2| dx = ∫₋₂⁵ (x+2) dx = [x²/2 + 2x]₋₂⁵ = [(25/2) + 10] - [(4/2) + (-4)] = (25/2 + 10) - (2 - 4) = (25/2 + 10) - (-2) = 25/2 + 12 = 25/2 + 24/2 = 49/2</p><p><strong>Step 4:</strong> Add both parts: 21/2 + 49/2 = 70/2 = 35</p><p><em>Note: Rechecking Step 2 more carefully: ∫₋₅⁻² (-x-2) dx = [-x²/2 - 2x]₋₅⁻² = (-2 + 4) - (-25/2 + 10) = 2 - (-5/2) = 9/2. Then 9/2 + 49/2 = 58/2 = 29</em></p><p>∴ Answer: <strong>29</strong></p>
Correct Answer: 29