>>219,220
u=x^3
x=u^(1/3)
dx=(1/3)u^(-2/3)du
∫[0,∞]1/√(x^3+1) dx=∫[0,∞](1/3)(u+1)^(-1/2)u^(-2/3)du
u=t/(1-t)
du=dt/(1-t)^2
∫[0,∞](1/3)(u+1)^(-1/2)u^(-2/3)du=∫[0,1](1/3)(1-t)^(1/2)t^(-2/3)(1-t)^(2/3)dt/(1-t)^2
=∫[0,1](1/3)(1-t)^(-5/6)t^(-2/3)dt
=(1/3)B(1/6,1/3)
=(1/3)Γ(1/6)Γ(1/3)/Γ(1/2)
Γ(1/3)=Γ(2/6)=２^(-2/3)Γ(1/6)Γ(2/3)/√π
Γ(1/3)Γ(2/3)=π/sin(π/3)=2π/√3
Γ(1/2)=√π
(1/3)Γ(1/6)Γ(1/3)/Γ(1/2)=Γ(1/3)^3/(3^(1/2)2^(1/3)π)