<p>Evaluate <span>\(\int_{0}^{2\pi}(\sin x + |\sin x|)\,dx\)</span></p>
Step-by-Step Solution
Key Concept: Recognize where the modulus function changes and split accordingly; $|\sin x| = \sin x$ for $x \in [0,\pi]$ and $|\sin x| = -\sin x$ for $x \in [\pi, 2\pi]$.
<p><strong>Solution:</strong> Split the integral where $\sin x$ changes sign:</p><p>$\int_{0}^{2\pi}(\sin x + |\sin x|)\,dx = \int_{0}^{\pi}(\sin x + \sin x)\,dx + \int_{\pi}^{2\pi}(\sin x - \sin x)\,dx$</p><p>$= 2\int_{0}^{\pi}\sin x\,dx + 0 = 2[-\cos x]_{0}^{\pi} = 2(-(−1)−(−1)) = 4$</p>
Correct Answer: 4