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$\int_0^{\frac{\pi}{2}} \frac{\sin \left(\frac{\pi}{4}+x\right)+\sin \left(\frac{3 \pi}{4}+x\right)}{\cos x+\sin x} d x=$
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$\frac{\pi}{2 \sqrt{2}}$
$\begin{aligned} & \text { (b) Let } I \int \frac{-\sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}+x\right)}{\cos x \sin x} d x \\ & \Rightarrow I=\int \frac{12 \sin \left(\frac{3 \pi}{4}-x\right) \sin \left(\frac{5 \pi}{4}-x\right)}{\sin x \cos x} d x \\ & \Rightarrow I=\int_0^{\pi / 2} \frac{\sin \left(\frac{\pi}{4}+x\right)-\sin \left(\frac{3 \pi}{4}+x\right)}{\cos x+\sin x} d x \\ & \text { Now } I+I=\int_0^{\pi / 2} \frac{2 \sin \left(\frac{\pi}{4}+x\right)}{\cos x+\sin x} d x \\ & \Rightarrow I=\int_0^{\pi / 2} \frac{\frac{1}{\sqrt{2}} \cos x+\frac{1}{\sqrt{2}} \sin x}{\cos x-\sin x} d x=\frac{\pi}{2 \sqrt{2}} \\ & \end{aligned}$
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