int_0^{pi /2} frac{sin x}{sin x + cos x} , dx | vedclass

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(C) Let $I = \int_0^{\pi /2} \frac{\sin x}{\sin x + \cos x} \, dx$  --- $(1)$Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$,we get:$

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$\frac{\pi }{4}$

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(C) Let $I = \int_0^{\pi /2} \frac{\sin x}{\sin x + \cos x} \, dx$  --- $(1)$Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$,we get:$I = \int_0^{\pi /2} \frac{\sin(\pi/2 - x)}{\sin(\pi/2 - x) + \cos(\pi/2 - x)} \, dx$$I = \int_0^{\pi /2} \frac{\cos x}{\cos x + \sin x} \, dx$ --- $(2)$Adding $(1)$ and $(2)$:$2I = \int_0^{\pi /2} \frac{\sin x + \cos x}{\sin x + \cos x} \, dx$$2I = \int_0^{\pi /2} 1 \, dx$$2I = [x]_0^{\pi /2} = \frac{\pi}{2} - 0 = \frac{\pi}{2}$$I = \frac{\pi}{4}$

$\int_0^{\pi /2} \frac{\sin x}{\sin x + \cos x} \, dx$ equals

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