find-J-n-p-0-x-n-e-x-2-p-dx-with-p-gt-0-and-n-integr-

Question Number 34910 by abdo imad last updated on 12/May/18

f i n d J_{n, p} = \int_{0}^{\infty} x^{n} e^{- \frac{x^{2}}{p}} d x w i t h p > 0 a n d n i n t e g r

Commented by candre last updated on 13/May/18

u = x^{n - 1} \Rightarrow d u = (n - 1) x^{n - 2} d x

d v = {x e}^{- \frac{x^{2}}{p}} d x \Rightarrow v = \int {x e}^{- \frac{x^{2}}{p}} d x

w = - \frac{x^{2}}{p} \Rightarrow d w = - \frac{2 x}{p} d x

v = - \frac{p}{2} \int e^{w} d w = - \frac{p}{2} e^{- \frac{x^{2}}{p}}

\underset{0}{\int^{\infty}} x^{n} e^{- \frac{x^{2}}{p}} d x = - {[\frac{{p x}^{n - 1} e^{- \frac{x^{2}}{p}}}{2}]}_{0}^{\infty} + \frac{(n - 1) p}{2} \underset{0}{\int^{\infty}} x^{n - 2} e^{- \frac{x^{2}}{p}} d x

J_{n, p} = \frac{(n - 1) {p J}_{n - 2, p}}{2}

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