$\displaystyle\int_0^{\pi/2} \dfrac{\sin^4 x}{\sin^4 x+\cos^4 x}\,dx$ equals

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_