Prove that ∫ dx/(1+cot^3x)=π/4, x ∈[0,π/2]

Question

Prove that  

\(\int\limits_0^{\pi/2}\cfrac{dx}{(1+cot^3x)}=\cfrac{\pi}{4}\)

∫ dx/(1+cot³x)=π/4, x  ∈[0,π/2]

Answer 1

\(y=\int\limits_0^{\pi/2}\cfrac{dx}{(1+\frac{cos^3x}{sin^3x})}dx\)

\(y=\int\limits_0^{\pi/2}\cfrac{sin^3x}{sin^3x+cos^3x}dx...(1)\)

Use King theorem of definite integral