The value of the integral $I = \int_{\pi}^{\pi} (|\sin x| + |\cos x|)dx$ (where $[\cdot]$ denotes the greatest integer function) is equal to
Step-by-Step Solution
Key Concept: Recognizing the geometric interpretation of the integrand as a constant function over the integration interval
Let $y = |\sin x| + |\cos x|$, so $y^2 = 1 + |\sin(2x)| \in [1,2]$, giving $y \in [1, \sqrt{2}]$. Therefore $\int_0^\pi 1\,dx = \pi - 0 = \pi$.
Correct Answer: π