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 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

Commented bytanmay.chaudhury50@gmail.com last updated on 28/Apr/18

do you like the different method to find the solution  ...it is not available in apps..

$${do}\:{you}\:{like}\:{the}\:{different}\:{method}\:{to}\:{find}\:{the}\:{solution} \\ $$ $$...{it}\:{is}\:{not}\:{available}\:{in}\:{apps}.. \\ $$

Commented byNECx last updated on 28/Apr/18

which app is this please?

$${which}\:{app}\:{is}\:{this}\:{please}? \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18

Terms of Service

Privacy Policy

Contact: info@tinkutara.com