find-I-n-p-0-x-n-e-px-with-n-and-p-from-N-

Question Number 30527 by abdo imad last updated on 22/Feb/18

f i n d I_{n, p} = \int_{0}^{\infty} x^{n} e^{- p x} w i t h n a n d p f r o m N^{★} .

Answered by sma3l2996 last updated on 23/Feb/18

I = \int_{0}^{\infty} x^{n} e^{- p x} d x

u = x^{n} \Rightarrow u^{'} = {n x}^{n - 1}

v^{'} = e^{- p x} \Rightarrow v = \frac{- 1}{p} e^{- p x}

I = - \frac{1}{p} {[x^{n} e^{- p x}]}_{0}^{\infty} + \frac{n}{p} \int_{0}^{\infty} x^{n - 1} e^{- p x} d x ({\underset{x \to \infty}{l i m x}}^{n} e^{- p x} = 0)

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

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

v^{'} = e^{- p x} \Rightarrow v = \frac{- 1}{p} e^{- p x}

I = \frac{n (n - 1)}{p^{2}} \int_{0}^{\infty} x^{n - 2} e^{- p x} d x

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k t i m e s

I = \frac{n (n - 1) (n - 2) \dots (n - k + 1)}{p^{k}} \int_{0}^{\infty} x^{n - k} e^{- p x} d x

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n t i m e s

I = \frac{n (n - 1) (n - 2) \dots (n - n + 1)}{p^{n}} \int_{0}^{\infty} e^{- p x} d x

I = \frac{n!}{p^{n}} {[- \frac{1}{p} e^{- p x}]}_{0}^{\infty} = \frac{n!}{p^{n + 1}} (- ({\underset{x \to \infty}{l i m e}}^{- p x} - e^{0}))

I = \frac{n!}{p^{n + 1}}