dx-1-x-

Question Number 80515 by M±th+et£s last updated on 03/Feb/20

\int \frac{d x}{{(1 + x^{ϕ})}^{ϕ}}

Commented by john santu last updated on 04/Feb/20

t = 1 + x^{ϕ} \Rightarrow x = {(t - 1)}^{\frac{1}{ϕ}}

I = \int_{0}^{\infty} \frac{1}{t^{ϕ}} . \frac{1}{ϕ} {(t - 1)}^{\frac{1}{ϕ} - 1} d t

= \frac{1}{ϕ} \int_{0}^{\infty} t^{- ϕ + ϕ^{- 1}} {(1 - t^{- 1})}^{\frac{1}{ϕ} - 1} d t

= \frac{1}{ϕ} . \frac{Γ (ϕ - \frac{1}{ϕ}) Γ (\frac{1}{ϕ})}{Γ (ϕ)}

= \frac{1}{ϕ} . \frac{Γ (ϕ^{- 1})}{ϕ^{- 1} Γ (ϕ^{- 1)})} = 1

w h e r e ϕ = \frac{\sqrt{5} - 1}{2}