Calculate $\int_{0}^{\frac{\pi}{2}}\frac{\cos(x)^{\sin(x)}}{(\cos x)^{\sin(x)}+(\sin x)^{\cos(x)}}dx$

Question

Calculate $$∫_{0}^{\frac{π}{2}}\frac{\cos \left(x\right)^{\sin \left(x\right)}}{\left(\cos x\right)^{\sin \left(x\right)}+\left(\sin x\right)^{\cos \left(x\right)}}dx$$.

EDIT: By changing the variable, $x→ \frac{π}{2}-x$,

$$∫_{0}^{\frac{π}{2}}\frac{\cos \left(x\right)^{\sin \left(x\right)}}{\left(\cos y\right)^{\sin \left(x\right)}+\left(\sin x\right)^{\cos x}}dx=∫_{0}^{\frac{π}{2}}\frac{\left(\sin x\right)^{\cos x}}{\left(\cos x\right)^{\sin \left(x\right)}+\left(\sin x\right)^{\cos y}}dx$$

Answer 1

$$\int_{0}^{\frac{\pi}{2}}\frac{(\cos{(x)})^{\sin{(x)}}}{(\cos{(x)})^{\sin{(x)}}+(\sin{(x)})^{\cos{(x)}}}dx$$ 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 $$\int_{0}^{\frac{\pi}{2}}\frac{(\cos{(x)})^{\sin{(x)}}}{(\cos{(x)})^{\sin{(x)}}+(\sin{(x)})^{\cos{(x)}}}dx=\frac{\pi}{4}$$