∫(sin 2x)/(sin⁴x + cos⁴x) dx equals:

∫(sin 2x)/(sin⁴x + cos⁴x) dx equals:

• A. tan⁻¹(tan²x) + C
• B. cot⁻¹(tan²x) + C
• C. Both 0 and 1
• D. tan⁻¹(2tan x) + C

Answer: C) Both 0 and 1

Explanation: sin 2x = 2sin x cos x. Denominator = (sin²x)² + (cos²x)² = (sin²x+cos²x)² − 2sin²x cos²x = 1 − (sin²2x)/2. Substitute u = sin²x, du = sin 2x dx. Integral becomes ∫du/(1 − 2u + 2u²) = tan⁻¹(2u−1) or tan⁻¹(tan²x). Both options 0 and 1 refer to the same family.