<p>Evaluate \(\displaystyle\int_0^{\pi/2}\frac{\sin x-\cos x}{1+\sin x\cos x}\,dx\) [JEE Main 2016]</p>
Step-by-Step Solution
Key Concept: By King's rule (x\to \pi/2-x), numerator changes sign while denominator is symmetric. So I = -I \to I = 0.
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<p>Let $I=\int_0^{\pi/2}\frac{\sin x-\cos x}{1+\sin x\cos x}dx$. King ($x\to\pi/2-x$):</p>
<p>Numerator: $\sin(\pi/2-x)-\cos(\pi/2-x)=\cos x-\sin x=-(\sin x-\cos x)$</p>
<p>Denominator: $1+\cos x\sin x$ (same). So $I=-I\Rightarrow \boxed{I=0}$.</p>
Correct Answer: A