Question Number 31512 by abdo imad last updated on 09/Mar/18
$${find}\:{lim}_{{x}\rightarrow\infty} \:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}\:{dt}. \\ $$
Commented by abdo imad last updated on 12/Mar/18
![∃ c ∈]x,2x[ / ∫_x ^(2x) ((cos((1/t)))/t)dt=cos((1/c))∫_x ^(2x) (dt/t)=cos((1/c))ln2 so lim_(x→∞) ∫_x ^(2x) ((cos((1/t)))/t)dt =lim_(c→∞) cos((1/c))ln2=ln2 .](https://www.tinkutara.com/question/Q31697.png)
$$\left.\exists\:{c}\:\in\right]{x},\mathrm{2}{x}\left[\:/\:\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}{dt}={cos}\left(\frac{\mathrm{1}}{{c}}\right)\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{dt}}{{t}}={cos}\left(\frac{\mathrm{1}}{{c}}\right){ln}\mathrm{2}\:{so}\right. \\ $$$${lim}_{{x}\rightarrow\infty} \:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}{dt}\:={lim}_{{c}\rightarrow\infty} {cos}\left(\frac{\mathrm{1}}{{c}}\right){ln}\mathrm{2}={ln}\mathrm{2}\:. \\ $$