Menu Close

0-y-a-1-1-y-b-dy-u-1-1-y-0-1-1-u-a-1-u-a-1-u-b-du-u-2-0-1-u-b-a-1-1-u-a-1-du-




Question Number 173975 by savitar last updated on 22/Jul/22
            ∫_0 ^∞ (y^(a−1) /((1+y)^b )) dy =^(u=(1/(1+y)))    ∫_0 ^1 (((1−u)^(a−1) )/u^(a−1) ) u^b   (du/u^2 )                                        = ∫_0 ^1  u^(b−a−1) (1−u)^(a−1) du                                       = B(b−a,a)=((Γ(b−a)Γ(a))/(Γ(b)))
$$ \\ $$$$\: \\ $$$$ \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{{y}^{{a}−\mathrm{1}} }{\left(\mathrm{1}+{y}\right)^{{b}} }\:{dy}\:\overset{{u}=\frac{\mathrm{1}}{\mathrm{1}+{y}}} {=}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left(\mathrm{1}−{u}\right)^{{a}−\mathrm{1}} }{{u}^{{a}−\mathrm{1}} }\:{u}^{{b}} \:\:\frac{{du}}{{u}^{\mathrm{2}} }\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{u}^{{b}−{a}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{a}−\mathrm{1}} {du} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{B}\left({b}−{a},{a}\right)=\frac{\Gamma\left({b}−{a}\right)\Gamma\left({a}\right)}{\Gamma\left({b}\right)} \\ $$$$\:\: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *