Question Number 86578 by Omer Alattas last updated on 29/Mar/20
Commented by john santu last updated on 29/Mar/20
![let F(x) antiderivative f(x) = ∫ _3 ^( x) ((sin t dt)/t) such that f(x) = F(x) −F(3) now lim_(x→3) ((x(F(x)−F(3)))/(x−3)) = 3× lim_(x→3) ((F(x)−F(3))/(x−3)) [ L′hopital rule ] 3× F′(3) = 3× ((sin (3))/3) = sin (3)](https://www.tinkutara.com/question/Q86583.png)
$$\mathrm{let}\:\mathrm{F}\left(\mathrm{x}\right)\:\mathrm{antiderivative}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\int\underset{\mathrm{3}} {\overset{\:\mathrm{x}} {\:}}\:\frac{\mathrm{sin}\:\mathrm{t}\:\mathrm{dt}}{\mathrm{t}} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{F}\left(\mathrm{x}\right)\:−\mathrm{F}\left(\mathrm{3}\right) \\ $$$$\mathrm{now}\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\mathrm{x}\left(\mathrm{F}\left(\mathrm{x}\right)−\mathrm{F}\left(\mathrm{3}\right)\right)}{\mathrm{x}−\mathrm{3}}\:=\: \\ $$$$\mathrm{3}×\:\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\:\frac{\mathrm{F}\left(\mathrm{x}\right)−\mathrm{F}\left(\mathrm{3}\right)}{\mathrm{x}−\mathrm{3}}\:\:\left[\:\mathrm{L}'\mathrm{hopital}\:\mathrm{rule}\:\right] \\ $$$$\mathrm{3}×\:\mathrm{F}'\left(\mathrm{3}\right)\:=\:\mathrm{3}×\:\frac{\mathrm{sin}\:\left(\mathrm{3}\right)}{\mathrm{3}}\:=\:\mathrm{sin}\:\left(\mathrm{3}\right) \\ $$
Commented by Omer Alattas last updated on 29/Mar/20
$${thnks} \\ $$