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Find-0-x-e-x-dx-




Question Number 196515 by hardmath last updated on 26/Aug/23
Find:   ∫_0 ^( +∞)  x^𝛑  e^(−x)  dx = ?
$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:+\infty} \:\mathrm{x}^{\boldsymbol{\pi}} \:\mathrm{e}^{−\boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:=\:? \\ $$
Answered by aleks041103 last updated on 26/Aug/23
∫_0 ^∞ x^s e^(−x) dx=Γ(s+1)  ⇒∫_0 ^( ∞) x^π e^(−x) dx = Γ(π+1)
$$\int_{\mathrm{0}} ^{\infty} {x}^{{s}} {e}^{−{x}} {dx}=\Gamma\left({s}+\mathrm{1}\right) \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\:\infty} {x}^{\pi} {e}^{−{x}} {dx}\:=\:\Gamma\left(\pi+\mathrm{1}\right) \\ $$
Commented by aleks041103 last updated on 26/Aug/23
No normal answer
$${No}\:{normal}\:{answer} \\ $$
Commented by hardmath last updated on 26/Aug/23
thank you dear professor
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$

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