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

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Question

(A) ધારો કે $I = \int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x} + (\sin x)^{\cos x}} dx$  --- $(1)$ગુણધર્મ $\int_0^a f(x) dx = \int_0^a f

Vedclass · Accepted Answer

$\pi / 4$

Vedclass · Answer

(A) ધારો કે $I = \int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x} + (\sin x)^{\cos x}} dx$  --- $(1)$ગુણધર્મ $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ નો ઉપયોગ કરતા:$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$$I = \int_0^{\pi / 2} \frac{(\sin x)^{\cos x}}{(\sin x)^{\cos x} + (\cos x)^{\sin x}} dx$  --- $(2)$$(1)$ અને $(2)$ નો સરવાળો કરતા:$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$

$\int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x}+(\sin x)^{\cos x}} d x$ નું મૂલ્ય શું છે?

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