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Given $\displaystyle\int_0^{\pi/2} \dfrac{\sin^n x}{\sin^n x + \cos^n x}\, dx$, the value is:
ADepends on $n$
B$\pi/4$ for all $n$
C$\pi/2$ for all $n$
D$0$
Answer & Solution
Correct answer: B. $\pi/4$ for all $n$
By King's $x \to \pi/2 - x$, integrand swaps to $\cos^n x/(\cos^n x + \sin^n x)$. Adding to original: $2I = \int_0^{\pi/2} 1\, dx = \pi/2$, so $I = \pi/4$ regardless of $n$. Beautiful invariance — a recurring JEE Adv test.
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