Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

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.

Terms of Service

Privacy Policy

Contact: info@tinkutara.com