Question

Evaluate \[ \int_0^{\pi/2} \frac{\sin^4x}{\sin^4x + \cos^4x} \, dx \]

Hint:

In integrals of the form $\int_0^{\pi/2} f(\sin x,\cos x) dx$, use the property $I=\int_0^{\pi/2} f(\cos x,\sin x) dx$.

Answer 1

Step 1: Use property of definite integrals.
\[ I = \int_0^{\pi/2} \frac{\sin^4x}{\sin^4x+\cos^4x} dx \] Now, \[ I = \int_0^{\pi/2} \frac{\cos^4x}{\cos^4x+\sin^4x} dx \text{(by } x \to \tfrac{\pi}{2}-x \text{)} \]

Step 2: Add the two results.
\[ 2I = \int_0^{\pi/2} \frac{\sin^4x}{\sin^4x+\cos^4x} dx + \int_0^{\pi/2} \frac{\cos^4x}{\cos^4x+\sin^4x} dx \] \[ 2I = \int_0^{\pi/2} 1 \, dx = \frac{\pi}{2} \]

Step 3: Final value.
\[ I = \frac{\pi}{4} \]

Final Answer: \[ \boxed{\tfrac{\pi}{4}} \]