Mathematics

The value of $$\int _{ 0 }^{ \pi /2 }{ \cfrac { \sin { x } -\cos { x }  }{ 1+\sin { x } \cos { x }  }  } dx$$ is ____


ANSWER

$$0$$


SOLUTION
Let 
$$I=\displaystyle \int_{0}^{\pi/2}{\dfrac{\sin x-\cos x dx}{1+\sin x\cos x}}-----(1)$$
using $$\displaystyle \int_{a}^{b}{f(x)dx}=\displaystyle \int_{a}^{b}{f(a+b-x)dx}$$
$$I=\displaystyle \int_{0}^{\pi/2}{\dfrac{\cos x-\sin x dx}{1+\sin x\cos x}}-----(2)$$
$$(1)+(2)$$
$$2I=\displaystyle \int_{0}^{\pi/2}{\dfrac{\sin x-\cos x+\cos x-\sin x}{1+\sin x\cos x}}dx$$
$$2I=0\Rightarrow I=0$$
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Single Correct Medium Published on 17th 09, 2020
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