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Question Number 69328 by mhmd last updated on 22/Sep/19

$$\underset{0}{\stackrel{\pi /2}{\int }}\frac{\varphi \left(x\right)}{\varphi \left(x\right)+\varphi (\frac{\pi }{2}-x)}dx=$$

Commented by mathmax by abdo last updated on 22/Sep/19

$$letI={\int }_0^{\frac{\pi }{2}}\frac{\phi \left(x\right)}{\phi \left(x\right)+\phi (\frac{\pi }{2}-x)}dxchangement\frac{\pi }{2}-x=tgive$$$$I=-{\int }_0^{\frac{\pi }{2}}\frac{\phi (\frac{\pi }{2}-t)}{\phi (\frac{\pi }{2}-t)+\phi \left(t\right)}(-dt)={\int }_0^{\frac{\pi }{2}}\frac{\phi (\frac{\pi }{2}-x)}{\phi \left(x\right)+\phi (\frac{\pi }{2}-x)}\Rightarrow$$$$2I={\int }_0^{\frac{\pi }{2}}\frac{\phi \left(x\right)+\phi (\frac{\pi }{2}-x)}{\phi \left(x\right)+\phi (\frac{\pi }{2}-x)}dx={\int }_0^{\frac{\pi }{2}}dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}$$

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