<p>Evaluate \(\displaystyle\int_0^{\pi/2}\frac{dx}{1+\tan^3 x}\) [JEE Main 2017]</p>
Step-by-Step Solution
Key Concept: 1/(1+tan^3x) + 1/(1+cot^3x) = 1. By King's rule, I = \intcosx^3/(sinx^3+cosx^3)dx and I+I = \pi/2.
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<p>Note \(\dfrac{1}{1+\tan^3 x}=\dfrac{\cos^3 x}{\cos^3 x+\sin^3 x}\).</p>
<p>By King (\(x\to\pi/2-x\)): \(I=\int_0^{\pi/2}\dfrac{\sin^3 x}{\sin^3 x+\cos^3 x}dx\).</p>
<p>Add: \(2I=\int_0^{\pi/2}1\,dx=\dfrac{\pi}{2}\Rightarrow I=\boxed{\dfrac{\pi}{4}}\).</p>
Correct Answer: A