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Question Number 100216 by Rio Michael last updated on 25/Jun/20

if I = \int_{0}^{\frac{π}{2}} \frac{\sin x}{\sin x + \cos x} d x = \int_{0}^{\frac{π}{2}} \frac{\cos x}{\sin x + \cos x} d x

then I = ? ?

Commented by Dwaipayan Shikari last updated on 25/Jun/20

\frac{π}{4} (I think it should be)

Commented by Dwaipayan Shikari last updated on 25/Jun/20

\int_{0}^{\frac{π}{2}} \frac{sinx}{sinx + cosx} dx = \int_{0}^{\frac{π}{2}} \frac{\sin (\frac{π}{2} - x)}{\sin (\frac{π}{2} - x) + \cos (\frac{π}{2} - x)} dx = \int_{0}^{\frac{π}{2}} \frac{cosx}{sinx + cosx} dx = I

so, 2 I = \int_{0}^{\frac{π}{2}} \frac{sinx + cosx}{sinx + cosx} dx = \frac{π}{2}

So, I = \frac{π}{4}

Answered by Ar Brandon last updated on 25/Jun/20

{\begin{cases} I = \int_{0}^{\frac{π}{2}} \frac{sinx}{sinx + cosx} dx \dots \dots \dots . (1) \\ I = \int_{0}^{\frac{π}{2}} \frac{cosx}{sinx + cosx} dx \dots \dots \dots . . (2) \end{cases} \Rightarrow I + I = \int_{0}^{\frac{π}{2}} \frac{sinx + cosx}{sinx + cosx} dx

\Rightarrow 2 I = \int_{0}^{\frac{π}{2}} dx = \frac{π}{2} \Rightarrow I = \frac{π}{4}

Commented by Coronavirus last updated on 25/Jun/20

c l e a n

Commented by Ar Brandon last updated on 25/Jun/20

Ouais

Commented by Rio Michael last updated on 26/Jun/20

thank {you}^{'} ll