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find-0-ln-1-t-2-t-2-dt-




Question Number 32478 by prof Abdo imad last updated on 25/Mar/18
find  ∫_0 ^∞  ln(((1+t^2 )/t^2 ))dt
$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:{ln}\left(\frac{\mathrm{1}+{t}^{\mathrm{2}} }{{t}^{\mathrm{2}} }\right){dt} \\ $$
Commented by prof Abdo imad last updated on 30/Mar/18
I = ∫_0 ^(+∞)  ln(1+(1/t^2 ))dt  by parts  I = ][t ln(1+(1/t^2 ))]_0 ^(+∞)   − ∫_0 ^(+∞)  t  ((−2)/t^3 ) (1+(1/t^2 ))^(−1) dt  =  2 ∫_0 ^∞    (1/t^2 )  . (1/(1+(1/t^2 )))dt = 2 ∫_0 ^∞   (dt/(t^2  +1)) = 2(π/2)  =π  let prove that lim_(t→+∞)  tln(1+(1/t^2 ))=0  = lim_(u →0)  (1/u) ln (1+u^2 ) =lim_(u→∞)  u ((ln(1+u^2 ))/u^2 ) =0  also lim_(t→o) t ln(1 +(1/t^2 )) =lim_(t→0)  tln(1+t^2 ) −2tlnt  =0  finally   I  =π
$${I}\:=\:\int_{\mathrm{0}} ^{+\infty} \:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right){dt}\:\:{by}\:{parts} \\ $$$$\left.{I}\:=\:\right]\left[{t}\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)\right]_{\mathrm{0}} ^{+\infty} \:\:−\:\int_{\mathrm{0}} ^{+\infty} \:{t}\:\:\frac{−\mathrm{2}}{{t}^{\mathrm{3}} }\:\left(\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)^{−\mathrm{1}} {dt} \\ $$$$=\:\:\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\:\:.\:\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }}{dt}\:=\:\mathrm{2}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{1}}\:=\:\mathrm{2}\frac{\pi}{\mathrm{2}}\:\:=\pi \\ $$$${let}\:{prove}\:{that}\:{lim}_{{t}\rightarrow+\infty} \:{tln}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)=\mathrm{0} \\ $$$$=\:{lim}_{{u}\:\rightarrow\mathrm{0}} \:\frac{\mathrm{1}}{{u}}\:{ln}\:\left(\mathrm{1}+{u}^{\mathrm{2}} \right)\:={lim}_{{u}\rightarrow\infty} \:{u}\:\frac{{ln}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}{{u}^{\mathrm{2}} }\:=\mathrm{0} \\ $$$${also}\:{lim}_{{t}\rightarrow{o}} {t}\:{ln}\left(\mathrm{1}\:+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)\:={lim}_{{t}\rightarrow\mathrm{0}} \:{tln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\:−\mathrm{2}{tlnt} \\ $$$$=\mathrm{0}\:\:{finally}\:\:\:{I}\:\:=\pi \\ $$$$ \\ $$

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