find-the-domain-of-definition-of-this-function-for-t-0-1-x-x-2x-1-lnt-dt-ptiCantor-

Question Number 196023 by pticantor last updated on 16/Aug/23

find the domain of definition of this function for t∈]0;1[ 𝛒(x)=∫_x ^(2x) (1/(lnt))dt ptiCantor

$${find}\:{the}\:{domain}\:{of}\:{definition}\:{of}\:{this} \\ $$$$\left.{function}\:{for}\:{t}\in\right]\mathrm{0};\mathrm{1}\left[\right. \\ $$$$\:\:\:\:\:\boldsymbol{\rho}\left({x}\right)=\int_{{x}} ^{\mathrm{2}{x}} \frac{\mathrm{1}}{{lnt}}{dt} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{ptiCantor} \\ $$

Answered by sniper237 last updated on 16/Aug/23

t→(1/(lnt)) is continue on ]0;1[ , and ]1;+∞[ p(x) exist if 2x , x are in the same interval ⇒ 0< x<1/2 or 1< x

$$\left.{t}\rightarrow\frac{\mathrm{1}}{{lnt}}\:\:{is}\:{continue}\:{on}\:\:\:\right]\mathrm{0};\mathrm{1}\left[\:,\:{and}\:\right]\mathrm{1};+\infty\left[\right. \\ $$$${p}\left({x}\right)\:{exist}\:\:{if}\:\:\:\mathrm{2}{x}\:,\:{x}\:{are}\:{in}\:{the}\:{same}\:{interval} \\ $$$$\:\:\Rightarrow\:\:\mathrm{0}<\:{x}<\mathrm{1}/\mathrm{2}\:\:\:{or}\:\:\:\mathrm{1}<\:{x}\: \\ $$$$\: \\ $$

Commented by York12 last updated on 16/Aug/23