If [bullet] denotes the greatest integer func | vedclass

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(D) Let $f(x) = |\sin x| + |\cos x|$.We know that $|\sin x| \geq \sin^2 x$ and $|\cos x| \geq \cos^2 x$ for all $x \in \mathbb{R}$.Adding these, we ge

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$2\pi$

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(D) Let $f(x) = |\sin x| + |\cos x|$.We know that $|\sin x| \geq \sin^2 x$ and $|\cos x| \geq \cos^2 x$ for all $x \in \mathbb{R}$.Adding these, we get $|\sin x| + |\cos x| \geq \sin^2 x + \cos^2 x = 1$.Also, the maximum value of $|\sin x| + |\cos x|$ occurs at $x = \frac{\pi}{4}$, where $|\sin \frac{\pi}{4}| + |\cos \frac{\pi}{4}| = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2} \approx 1.414$.Since $1 \leq |\sin x| + |\cos x| \leq \sqrt{2}$, the greatest integer function $[|\sin x| + |\cos x|]$ will always be equal to $1$ for all $x \in [0, 2\pi]$.Therefore, $\int_0^{2 \pi} [|\sin x| + |\cos x|] \, dx = \int_0^{2 \pi} 1 \, dx = [x]_0^{2 \pi} = 2\pi - 0 = 2\pi$.

If $[\bullet]$ denotes the greatest integer function, then evaluate $\int_0^{2 \pi} [|\sin x| + |\cos x|] \, dx$.

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