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1-2-0-1-ln-x-y-x-y-dx-dy-




Question Number 32484 by Eng.Firas last updated on 25/Mar/18
  ∫_1 ^2 ∫_0 ^1 ((ln(x+y))/((x+y))) dx dy
$$ \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({x}+{y}\right)}{\left({x}+{y}\right)}\:{dx}\:{dy} \\ $$
Commented by abdo imad last updated on 26/Mar/18
I =∫_1 ^2  ( ∫_0 ^1   ((ln(x+y))/(x+y))dx)dy  let put A= ∫_0 ^1   ((ln(x+y))/(x+y))dx  ch. x+y =t give  A = ∫_y ^(1+y)   ((lnt)/t) dt  =[(1/2)( lnt)^2 ]_y ^(1+y)   =(1/2) ( (ln(1+y)^2  −(lny)^2 ) ⇒  I = (1/2) ∫_1 ^2  (ln(1+y))^2 dy  − (1/2) ∫_1 ^2  ((lny))^2 dy  but  ch. lny =t give   ∫_1 ^2  (lny)^2  dy = ∫_0 ^(ln(2))  t^2  e^t  dt = [t^2 e^t ]_0 ^(ln(2))  −∫_0 ^(ln(2))  2t e^t dt  = 2(ln2)^2  −2 ( [t e^t ]_0 ^(ln(2))  −∫_0 ^(ln(2)) e^t dt)  =2(ln2)^2   −2 (2ln2 −1)= 2(ln2)^2  −4ln(2) +1 also  ch. ln(1+y) =t give  ∫_1 ^2   (ln(1+y))^2  dy = ∫_(ln2) ^(ln(3))  t^2  e^t dt  and we get the value of  this intevral by the same method.....
$${I}\:=\int_{\mathrm{1}} ^{\mathrm{2}} \:\left(\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({x}+{y}\right)}{{x}+{y}}{dx}\right){dy}\:\:{let}\:{put}\:{A}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left({x}+{y}\right)}{{x}+{y}}{dx} \\ $$$${ch}.\:{x}+{y}\:={t}\:{give} \\ $$$${A}\:=\:\int_{{y}} ^{\mathrm{1}+{y}} \:\:\frac{{lnt}}{{t}}\:{dt}\:\:=\left[\frac{\mathrm{1}}{\mathrm{2}}\left(\:{lnt}\right)^{\mathrm{2}} \right]_{{y}} ^{\mathrm{1}+{y}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:\left(\:\left({ln}\left(\mathrm{1}+{y}\right)^{\mathrm{2}} \:−\left({lny}\right)^{\mathrm{2}} \right)\:\Rightarrow\right. \\ $$$${I}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\left({ln}\left(\mathrm{1}+{y}\right)\right)^{\mathrm{2}} {dy}\:\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\left(\left({lny}\right)\right)^{\mathrm{2}} {dy}\:\:{but} \\ $$$${ch}.\:{lny}\:={t}\:{give}\: \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} \:\left({lny}\right)^{\mathrm{2}} \:{dy}\:=\:\int_{\mathrm{0}} ^{{ln}\left(\mathrm{2}\right)} \:{t}^{\mathrm{2}} \:{e}^{{t}} \:{dt}\:=\:\left[{t}^{\mathrm{2}} {e}^{{t}} \right]_{\mathrm{0}} ^{{ln}\left(\mathrm{2}\right)} \:−\int_{\mathrm{0}} ^{{ln}\left(\mathrm{2}\right)} \:\mathrm{2}{t}\:{e}^{{t}} {dt} \\ $$$$=\:\mathrm{2}\left({ln}\mathrm{2}\right)^{\mathrm{2}} \:−\mathrm{2}\:\left(\:\left[{t}\:{e}^{{t}} \right]_{\mathrm{0}} ^{{ln}\left(\mathrm{2}\right)} \:−\int_{\mathrm{0}} ^{{ln}\left(\mathrm{2}\right)} {e}^{{t}} {dt}\right) \\ $$$$=\mathrm{2}\left({ln}\mathrm{2}\right)^{\mathrm{2}} \:\:−\mathrm{2}\:\left(\mathrm{2}{ln}\mathrm{2}\:−\mathrm{1}\right)=\:\mathrm{2}\left({ln}\mathrm{2}\right)^{\mathrm{2}} \:−\mathrm{4}{ln}\left(\mathrm{2}\right)\:+\mathrm{1}\:{also} \\ $$$${ch}.\:{ln}\left(\mathrm{1}+{y}\right)\:={t}\:{give} \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\left({ln}\left(\mathrm{1}+{y}\right)\right)^{\mathrm{2}} \:{dy}\:=\:\int_{{ln}\mathrm{2}} ^{{ln}\left(\mathrm{3}\right)} \:{t}^{\mathrm{2}} \:{e}^{{t}} {dt}\:\:{and}\:{we}\:{get}\:{the}\:{value}\:{of} \\ $$$${this}\:{intevral}\:{by}\:{the}\:{same}\:{method}….. \\ $$
Commented by Eng.Firas last updated on 26/Mar/18
Thank you
$${Thank}\:{you} \\ $$

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