$\displaystyle\int_0^{\pi/2} \dfrac{\sin^4 x}{\sin^4 x+\cos^4 x}\,dx$ equals
A$\dfrac{\pi}{2}$
B$\dfrac{\pi}{4}$
C$1$
D$\dfrac{\pi}{8}$
Answer & Solution
Correct answer: B. $\dfrac{\pi}{4}$
1. Let $I$ be the integral. By $\mathbf{P_4}$, replace $x$ by $\tfrac{\pi}{2}-x$: numerator becomes $\cos^4 x$.
2. So $I=\int_0^{\pi/2}\dfrac{\cos^4 x}{\cos^4 x+\sin^4 x}\,dx$.
3. Add the two forms: $2I=\int_0^{\pi/2} 1\,dx=\dfrac{\pi}{2}$.
4. Hence $I=\dfrac{\pi}{4}$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.345_
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