Help to proof the Inequality, problem d)

Question

Please help to proof the Inequality, problem d)

Answer 1

sinθ≈θ-θ³/3! for small θ, so if θ=2x³ and 0≤x<1:

sin(2x³)≈2x³-8x⁹/6=2x³(1-2x⁶/3).

∛sin(2x³)≈∛(2x³(1-2x⁶/3))=(x∛2)(1-2x⁶/3)^⅓≈(x∛2)(1-2x⁶/9).

∫∛sin(2x³)≈∛2∫(x-2x⁷/9)=∛2[x²/2-x⁸/36]=(x²∛2/2)(1-x⁶/18). The upper limit is x<1, so x⁶<1 and the expression further approximates to x²∛2/2. The lower limit (given) is zero. The upper limit as x→1 of this expression is ∛2/2, making the integral≤∛2/2, QED.