The value of int_0^{pi / 2} frac{(cos x)^{sin | vedclass

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(A) Let $I = \int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x} + (\sin x)^{\cos x}} dx$  --- $(1)$Using the property $\int_0^a f(x) dx = \in

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$\pi / 4$

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(A) Let $I = \int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x} + (\sin x)^{\cos x}} dx$  --- $(1)$Using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$,we get:$I = \int_0^{\pi / 2} \frac{(\cos(\pi/2 - x))^{\sin(\pi/2 - x)}}{(\cos(\pi/2 - x))^{\sin(\pi/2 - x)} + (\sin(\pi/2 - x))^{\cos(\pi/2 - x)}} dx$$I = \int_0^{\pi / 2} \frac{(\sin x)^{\cos x}}{(\sin x)^{\cos x} + (\cos x)^{\sin x}} dx$  --- $(2)$Adding $(1)$ and $(2)$:$2I = \int_0^{\pi / 2} \frac{(\cos x)^{\sin x} + (\sin x)^{\cos x}}{(\cos x)^{\sin x} + (\sin x)^{\cos x}} dx$$2I = \int_0^{\pi / 2} 1 dx$$2I = [x]_0^{\pi / 2} = \pi / 2$$I = \pi / 4$

The value of $\int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x}+(\sin x)^{\cos x}} d x$ is

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