Given $\displaystyle\int_0^{\pi/2} \dfrac{\sin^n x}{\sin^n x + \cos^n …

Question

Given $\displaystyle\int_0^{\pi/2} \dfrac{\sin^n x}{\sin^n x + \cos^n x}\, dx$, the value is:

Accepted Answer

B. $\pi/4$ for all $n$

By King's $x 	o \pi/2 - x$, 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.

Answer

Depends on $n$

Answer

$\pi/2$ for all $n$

Answer

$0$

Given $\displaystyle\int_0^{\pi/2} \dfrac{\sin^n x}{\sin^n x + \cos^n x}\, dx$, the value is:

Keep practising Unit 6B: Definite Integration & Riemann Sums