calculate-0-e-x-n-dx-

Question Number 124920 by mathmax by abdo last updated on 07/Dec/20

calculate \int_{0}^{\infty} e^{- x^{n}} dx

Answered by Dwaipayan Shikari last updated on 07/Dec/20

\int_{0}^{\infty} e^{- x^{n}} d x x^{n} = u \Rightarrow {n x}^{n - 1} = \frac{d u}{d x}

\Rightarrow \frac{1}{n} \int_{0}^{\infty} x^{1 - n} e^{- u} d u = \frac{1}{n} \int_{0}^{\infty} u^{\frac{1 - n}{n}} e^{- u} d u = \frac{1}{n} Γ (\frac{1}{n}) = Γ (\frac{n + 1}{n})

Answered by mathmax by abdo last updated on 07/Dec/20

A_{n} = \int_{0}^{\infty} e^{- x^{n}} dx we do the changement x = t^{\frac{1}{n}} \Rightarrow

A_{n} = \int_{0}^{\infty} e^{- t} \times \frac{1}{n} t^{\frac{1}{n} - 1} dt = \frac{1}{n} \int_{0}^{\infty} t^{\frac{1}{n} - 1} e^{- t} dt

= \frac{1}{n} \times Γ (\frac{1}{n})