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find-0-x-2n-1-e-x-2-dx-with-n-from-N-

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Question Number 30761 by abdo imad last updated on 25/Feb/18
find ∫_0 ^∞ x^(2n+1) e^(−x^2 ) dx with n from N.
find∫0∞x2n+1e−x2dxwithnfromN.
Commented by abdo imad last updated on 28/Feb/18
let put A_n = ∫_0 ^∞ x^(2n+1) e^(−x^2 ) dx A_n = ((−1)/2) ∫_0 ^∞ x^(2n) (−2x e^(−x^2 ) )dx let ntegrate by parts u^′ =−2x e^(−x^2 ) and v=x^(2n) ⇒ A_n =−(1/2) ( [x^(2n) e^(−x^2 ) ]_0 ^∞ −∫_0 ^∞ 2n x^(2n−1) e^(−x^2 ) dx ) A_n =n A_(n−1) ⇒Π_(k=1) ^n A_k =n! Π_(k=1) ^n A_(k−1) ⇒ A_n =n! A_0 but A_0 = ∫_0 ^∞ x e^(−x^2 ) dx = [−(1/2)e^(−x^2 ) ]_0 ^∞ =(1/2) ⇒ A_n = ((n!)/2) .
letputAn=∫0∞x2n+1e−x2dxAn=−12∫0∞x2n(−2xe−x2)dxletntegratebypartsu′=−2xe−x2andv=x2n⇒An=−12([x2ne−x2]0∞−∫0∞2nx2n−1e−x2dx)An=nAn−1⇒∏k=1nAk=n!∏k=1nAk−1⇒An=n!A0butA0=∫0∞xe−x2dx=[−12e−x2]0∞=12⇒An=n!2.

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