Question

By using the properties of definite integrals,evaluate the integrals: $\int^\frac{\pi}{2}_0 {\frac{sin\ x- cos\ x}{1+ sin\ x\ cos\ x}} \, dx$

Answer 1

Answer:

0

Step-by-step explanation:

$concept:\\\\\int\limits^{\frac{\pi}{2}}_0{f(x)\:dx=\int\limits^{\frac{\pi}{2}}_0{f(\frac{\pi}{2}-x)\:dx$

$Let,\\\\I=\int\limits^{\frac{\pi}{2}}_0\frac{sinx-cosx}{1+sinx.cosx}\:dx........(1)$

By properties of definite integral,

$I=\int\limits^{\frac{\pi}{2}}_0\frac{sin(\frac{\pi}{2}-x)-cos(\frac{\pi}{2}-x)}{1+sin(\frac{\pi}{2}-x).cos(\frac{\pi}{2}-x)}\:dx\\\\I=\int\limits^{\frac{\pi}{2}}_0\frac{cosx-sinx}{1+cosx.sinx}\:dx\\\\I=-\int\limits^{\frac{\pi}{2}}_0\frac{sinx-cosx}{1+sinx.cosx}\:dx........(2)\\\\using (1) in (2)\\\\I=-\:I\\\\2I=0\\\\Hence, I=0$