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 154927 by physicstutes last updated on 23/Sep/21

Let I_n =∫x^n e^(−x) dx  show that   ∫_0 ^∞ x^n e^(−x) dx=n!

$$\mathrm{Let}\:{I}_{{n}} =\int{x}^{{n}} {e}^{−{x}} {dx} \\ $$$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{{n}} {e}^{−{x}} {dx}={n}! \\ $$

Answered by ArielVyny last updated on 23/Sep/21

use gamma function.  Γ(m)=∫_0 ^∞ e^(−x) x^(m−1) dx and we know that  Γ(m+1)=m!  then ∫_0 ^∞ x^n e^(−x) dx=Γ(n+1)=n!

$${use}\:{gamma}\:{function}. \\ $$$$\Gamma\left({m}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {x}^{{m}−\mathrm{1}} {dx}\:{and}\:{we}\:{know}\:{that} \\ $$$$\Gamma\left({m}+\mathrm{1}\right)={m}! \\ $$$${then}\:\int_{\mathrm{0}} ^{\infty} {x}^{{n}} {e}^{−{x}} {dx}=\Gamma\left({n}+\mathrm{1}\right)={n}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com