Menu Close

let-consider-x-x-x-1-prove-that-a-gt-0-0-1-a-x-dx-ln-a-2-prove-that-n-N-0-1-x-sin-2pinx-dx-pi-2-




Question Number 33897 by math khazana by abdo last updated on 26/Apr/18
let consider ψ(x)=((Γ^′ (x))/(Γ(x)))  1) prove that ∀ a>0  ∫_0 ^1 ψ(a+x)dx=ln(a)  2) prove that ∀ n∈ N^★  ∫_0 ^1  ψ(x)sin(2πnx)dx=−(π/2)
$${let}\:{consider}\:\psi\left({x}\right)=\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \psi\left({a}+{x}\right){dx}={ln}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\in\:{N}^{\bigstar} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\psi\left({x}\right){sin}\left(\mathrm{2}\pi{nx}\right){dx}=−\frac{\pi}{\mathrm{2}} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18
Answered by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18

Leave a Reply

Your email address will not be published.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