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Calculate $\int_{0}^{\frac{\pi}{2}}\frac{cos(x)^{sin(x)}}{(cosx)^{\sin(x)}+(sinx)^{cos(x)}}dx$. EDIT: By changing the variable, $x\rightarrow \frac{\pi}{2}-x$, $\int_{0}^{\frac{\pi}{2}}\frac{cos(x)^{sin(x)}}{(cosy)^{sin(x)}+(sinx)^{cosx}}dx=\int_{0}^{\frac{\pi}{2}}\frac{(sinx)^{cosx}}{(cosx)^{sin(x)}+(sinx)^{cosy}}dx$

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$$\int_{0}^{\frac{\pi}{2}}\frac{(\cos{(x)})^{\sin{(x)}}}{(\cos{(x)})^{\sin{(x)}}+(\sin{(x)})^{\cos{(x)}}}dx$$ Using instead the change of variable $x\to\frac{\pi}2 - x$ we have $dx\to-dx$ and then $$\int_{0}^{\frac{\pi}{2}}\frac{(\cos{(x)})^{\sin{(x)}}}{(\cos{(x)})^{\sin{(x)}}+(\sin{(x)})^{\cos{(x)}}}dx=\int_0^{\frac{\pi}2}\frac{(\sin{(x)})^{\cos{(x)}}}{(\cos{(x)})^{\sin{(x)}}+(\sin{(x)})^{\cos{(x)}}}dx$$ Hence adding the two integrals gives the solution $$\int_{0}^{\frac{\pi}{2}}\frac{(\cos{(x)})^{\sin{(x)}}}{(\cos{(x)})^{\sin{(x)}}+(\sin{(x)})^{\cos{(x)}}}dx=\frac{\pi}{4}$$

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