Question:

<div class='solution'> <p>\(1-\sin 2x=\sin^2 x-2\sin x\cos x+\cos^2 x=(\sin x-\cos x)^2\)</p> <p>\(\sqrt{1-\sin 2x}=|\sin x-\cos x|\)</p> <p>On \([0,\pi/4]\): \(\cos x>\sin x\), so \(|\sin x-\cos x|=\cos x-\sin x\).</p> <p>On \([\pi/4,\pi/2]\): \(\sin x>\cos x\), so \(|\sin x-\cos x|=\sin x-\cos x\).</p> <p>\[I=\int_0^{\pi/4}(\cos x-\sin x)dx+\int_{\pi/4}^{\pi/2}(\sin x-\cos x)dx\]</p> <p>\(=[\sin x+\cos x]_0^{\pi/4}+[-\cos x-\sin x]_{\pi/4}^{\pi/2}\)</p> <p>\(=(\sqrt{2}-1)+(-1-(-\sqrt{2}))=(\sqrt{2}-1)+(\sqrt{2}-1)=2(\sqrt{2}-1)\)</p> </div>