Question Number 32044 by abdo imad last updated on 18/Mar/18
$${fimd}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:{dt}\:. \\ $$
Commented by abdo imad last updated on 20/Mar/18
$$\:\:{ln}\left(\mathrm{1}+{sint}\right)\:\sim\:{ln}\left(\mathrm{1}+{t}\right)\sim\:{t}\:\Rightarrow{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:\sim{t}^{\mathrm{3}} \:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right){dt}\:\:\sim\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{3}} {dt}\:=\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} {ln}\left(\mathrm{1}+{sint}\right){dt}\:\:={lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}}{\mathrm{4}}\:=\mathrm{0} \\ $$