Question

Ma ajuta si pe mine cineva va rog.$\int\limits^\frac{ \pi }{2} _0 { \frac{sinx}{sinx+cosx} } \, dx$

Answer 1

Salut,

$I=\int\limits^{\frac{\pi}2}_0\dfrac{sinx}{sinx+cosx}dx,\ J=\int\limits^{\frac{\pi}2}_0\dfrac{cosx}{sinx+cosx}dx; I+J=\int\limits^{\frac{\pi}2}_0dx=\dfrac{\pi}2;\\\\J-I=\int\limits^{\frac{\pi}2}_0\dfrac{cosx-sinx}{sinx+cosx}dx=\int\limits^{\frac{\pi}2}_0\left(ln|sinx+cosx|\right)'dx=\\\\=ln\left|sin\left(\dfrac{\pi}2\right)+cos\left(\dfrac{\pi}2\right)\right|-ln|sin0+cos0|=ln|1|-ln|1|=0.$

Am obținut că I = J, și I+J = π/2, deci I = π/4. 

Green eyes.