Question

Evaluate: \[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \, dx \]

Hint:

Use the identity \(\sin 2x = 2 \sin x \cos x\) to simplify trigonometric expressions in integrals involving products of sine and cosine.

Answer 1

First, recall the identity: \[ \sin 2x = 2 \sin x \cos x \] Thus, \[ \sin x \cos x = \frac{1}{2} \sin 2x \] The integral becomes: \[ I = \int_0^{\frac{\pi}{4}} \frac{\frac{1}{2} \sin 2x}{\cos^4 x + \sin^4 x} \, dx \] Now, let’s simplify the denominator: \[ \cos^4 x + \sin^4 x = (\cos^2 x + \sin^2 x)^2 - 2\cos^2 x \sin^2 x \] Since \(\cos^2 x + \sin^2 x = 1\), we get: \[ \cos^4 x + \sin^4 x = 1 - 2 \cos^2 x \sin^2 x \] Now, substitute this into the integral: \[ I = \int_0^{\frac{\pi}{4}} \frac{\frac{1}{2} \sin 2x}{1 - 2 \cos^2 x \sin^2 x} \, dx \] This can be solved using standard methods or numerical techniques. The final result can be obtained after solving the integral.