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fimd-lim-x-0-1-x-3-0-x-t-2-ln-1-sint-dt-




Question Number 32044 by abdo imad last updated on 18/Mar/18
fimd  lim_(x→0)      (1/x^3 ) ∫_0 ^x  t^2  ln(1+sint) dt .
$${fimd}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:{dt}\:. \\ $$
Commented by abdo imad last updated on 20/Mar/18
  ln(1+sint) ∼ ln(1+t)∼ t ⇒t^2  ln(1+sint) ∼t^3  ⇒  ∫_0 ^x  t^2  ln(1+sint)dt  ∼ ∫_0 ^x  t^3 dt =(x^4 /4) ⇒  lim_(x→0)  ∫_0 ^x  t^2 ln(1+sint)dt  =lim_(x→0)  (x/4) =0
$$\:\:{ln}\left(\mathrm{1}+{sint}\right)\:\sim\:{ln}\left(\mathrm{1}+{t}\right)\sim\:{t}\:\Rightarrow{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right)\:\sim{t}^{\mathrm{3}} \:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{ln}\left(\mathrm{1}+{sint}\right){dt}\:\:\sim\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{3}} {dt}\:=\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} {ln}\left(\mathrm{1}+{sint}\right){dt}\:\:={lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}}{\mathrm{4}}\:=\mathrm{0} \\ $$

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