find-0-e-x-3-dx-

Question Number 79527 by mathmax by abdo last updated on 25/Jan/20

f i n d \int_{0}^{\infty} e^{- x^{3}} d x

Commented by mathmax by abdo last updated on 26/Jan/20

l e t I = \int_{0}^{\infty} e^{- x^{3}} d x c h a n g e m e n t x^{3} = t g i v e x = t^{\frac{1}{3}}

I = \int_{0}^{\infty} e^{- t} \frac{1}{3} t^{\frac{1}{3} - 1} d t = \frac{1}{3} \int_{0}^{\infty} t^{\frac{1}{3} - 1} e^{- t} d t w e h a v e

Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t (x > 0) \Rightarrow I = \frac{1}{3} Γ (\frac{1}{3})