0-pi-2-x-x-pi-2-x-dx-

Question Number 69328 by mhmd last updated on 22/Sep/19

\underset{0}{\int^{π / 2}} \frac{ϕ (x)}{ϕ (x) + ϕ (\frac{π}{2} - x)} d x =

Commented by mathmax by abdo last updated on 22/Sep/19

l e t I = \int_{0}^{\frac{π}{2}} \frac{φ (x)}{φ (x) + φ (\frac{π}{2} - x)} d x c h a n g e m e n t \frac{π}{2} - x = t g i v e

I = - \int_{0}^{\frac{π}{2}} \frac{φ (\frac{π}{2} - t)}{φ (\frac{π}{2} - t) + φ (t)} (- d t) = \int_{0}^{\frac{π}{2}} \frac{φ (\frac{π}{2} - x)}{φ (x) + φ (\frac{π}{2} - x)} \Rightarrow

2 I = \int_{0}^{\frac{π}{2}} \frac{φ (x) + φ (\frac{π}{2} - x)}{φ (x) + φ (\frac{π}{2} - x)} d x = \int_{0}^{\frac{π}{2}} d x = \frac{π}{2} \Rightarrow I = \frac{π}{4}