find I_(n,p) = ∫_0 ^∞ x^n e^(−px) with n and p from N^★ .


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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

	$I = \int_{0}^{\infty} x^{n} e^{- p x} d x$ $u = x^{n} \Rightarrow u^{'} = {n x}^{n - 1}$ $v^{'} = e^{- p x} \Rightarrow v = \frac{- 1}{p} e^{- p x}$ $I = - \frac{1}{p} {[x^{n} e^{- p x}]}_{0}^{\infty} + \frac{n}{p} \int_{0}^{\infty} x^{n - 1} e^{- p x} d x ({\underset{x \to \infty}{l i m x}}^{n} e^{- p x} = 0)$ $= \frac{n}{p} \int_{0}^{\infty} x^{n - 1} e^{- p x} d x$ $u = x^{n - 1} \Rightarrow u^{'} = (n - 1) x^{n - 2}$ $v^{'} = e^{- p x} \Rightarrow v = \frac{- 1}{p} e^{- p x}$ $I = \frac{n (n - 1)}{p^{2}} \int_{0}^{\infty} x^{n - 2} e^{- p x} d x$ $.$ $.$ $.$ $k t i m e s$ $I = \frac{n (n - 1) (n - 2) . . . (n - k + 1)}{p^{k}} \int_{0}^{\infty} x^{n - k} e^{- p x} d x$ $.$ $.$ $.$ $n t i m e s$ $I = \frac{n (n - 1) (n - 2) . . . (n - n + 1)}{p^{n}} \int_{0}^{\infty} e^{- p x} d x$ $I = \frac{n!}{p^{n}} {[- \frac{1}{p} e^{- p x}]}_{0}^{\infty} = \frac{n!}{p^{n + 1}} (- ({\underset{x \to \infty}{l i m e}}^{- p x} - e^{0}))$ $I = \frac{n!}{p^{n + 1}}$
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