let-x-0-t-x-1-e-t-dt-with-x-gt-0-1-prove-that-x-1-x-pi-sin-pix-2-find-the-value-of-0-e-x-2-dx-

Question Number 33895 by math khazana by abdo last updated on 26/Apr/18

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) prove that Γ(x)Γ(1−x)= (π/(sin(πx))) 2) find the value of ∫_0 ^∞ e^(−x^2 ) dx .

$${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right)=\:\frac{\pi}{{sin}\left(\pi{x}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18

2.let t=x^2 dt/dx=2x =2(√t) ∫_0 ^∞ e^(−t) .1/2.t^(−1/2) dt =1/2∫_0 ^∞ e^(−t) t^(1/2−1) dt =1/2⌈(1/2) =1/2.(√Π)

$$\mathrm{2}.{let}\:{t}={x}^{\mathrm{2}} \\ $$$$\:\:\:\:{dt}/{dx}=\mathrm{2}{x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\sqrt{{t}} \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{e}^{−{t}} .\mathrm{1}/\mathrm{2}.{t}^{−\mathrm{1}/\mathrm{2}} \:{dt} \\ $$$$=\mathrm{1}/\mathrm{2}\underset{\mathrm{0}} {\overset{\infty} {\int}}{e}^{−{t}} {t}^{\mathrm{1}/\mathrm{2}−\mathrm{1}} {dt} \\ $$$$=\mathrm{1}/\mathrm{2}\lceil\left(\mathrm{1}/\mathrm{2}\right) \\ $$$$=\mathrm{1}/\mathrm{2}.\sqrt{\Pi} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18