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The value of $I=\int_0^{\pi / 2} \frac{(\sin x+\cos x)^2}{\sqrt{1+\sin 2 x}} d x$ is
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$\int_0^{\frac{\pi}{2}} \frac{(\sin x+\cos x)^2}{\sqrt{(\sin x+\cos x)^2}} d x=\int_0^{\frac{\pi}{2}}(\sin x+\cos x) d x=|-\cos x+\sin x|_0^{\frac{\pi}{2}}=2$
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