幫我檢查一下積分是哪部分出錯

題目: ∫ dx /(1 + x^4) 範圍是(0~infinit)

我的解法:
let x = (tanθ)^(1/2) ,then dx= (sec^2 θ) / [2(tanθ)^(1/2)]
,and cos^2 θ=1 /(1 + x^4)

則 ∫ dx /(1 + x^4) (0~infinit)
=∫ [(cotθ)^(1/2)]dθ/2 (0~pi/2)
=(1/4) * 2 ∫ [(sinθ)^(-1/2)][(cosθ)^(1/2)] dθ (0~pi/2)
=(1/4) * β(1/4,3/4)
=(1/4) * [Γ(1/4)Γ(3/4)]/Γ(1)

這裡使用一個Corollary,並已知Γ(1/2)=pi^(1/2)
Γ(x)=[2^(x-1)]*Γ(x/2)Γ((x+1)/2) / pi^(1/2)
=>Γ(1/2)=(2pi)^(-1/2) *Γ(1/4)Γ(3/4)
=>Γ(1/4)Γ(3/4) =2^(1/2) * pi

(1/4) * [Γ(1/4)Γ(3/4)]/Γ(1)
=(1/4)*2^(1/2) * pi
=(√2 *pi)/4


可是答案是(√2 *pi)/8