Question Number 167882 by LEKOUMA last updated on 28/Mar/22

Answered by ArielVyny last updated on 29/Mar/22
![t=(โ((1โx)/(1+x)))โdt=((โ(2/((1+x)^2 )))/(2t))dx=โ(1/((1+x)^2 ))(1/t)dx t^2 =((1โx)/(1+x))โ1โt^2 =x(t^2 +1)โx=((1โt^2 )/(1+t^2 )) dt=โ(1/((((1+t^2 )/(1+t^2 ))+((1โt^2 )/(1+t^2 )))t))dx=โ.((1+t^2 )/t)dx I=โโซ((1+t^2 )/(1โt^2 )).t.(t/(1+t^2 ))dt=โซ((โ1โt^2 +1)/(1โt^2 ))dt I=tโโซ(1/(1โt^2 ))dt t=sinuโdt=cosudu I=tโโซ(1/(cos^2 u))cosudu=(โ((1โx)/(1+x)))+โซ(1/(cosu))du I=(โ((1โx)/(1+x)))+โซ((cosu)/((1โsin^2 u)))du =.....โ(1/2)(โโซโ((cosu)/(1โsinu))+โซ((cosu)/(1+sinu))du) =.....โ(1/2)[โln(1โsinu)+ln(1+sinu)] =.....โ(1/2)ln(((1+sinu)/(1โsinu))) =.....โ(1/2)ln(((1+t)/(1โt)))=(โ(((1โx)/(1+x))โ))(1/2)ln(((1+(โ((1โx)/(1+x))))/(1โ(โ((1โx)/(1+x)))))) I=(โ(((1โx)/(1+x))โ))(1/2)ln(((((โ(1+x))+(โ(1โx)))/( (โ(1+x))))/(((โ(1+x))โ(โ(1โx)))/( (โ(1+x)))))) =(โ((1โx)/(1+x)))โ(1/2)ln((((โ(1+x))+(โ(1โx)))/( (โ(1+x))โ(โ(1โx))))) =(โ((1โx)/(1+x)))โ(1/2)ln[((((โ(1+x))+(โ(1โx)))^2 )/(((โ(1+x)))^2 โ((โ(1โx)))^2 ))] =(โ((1โx)/(1+x)))โ(1/2)ln[((1+(โ(1โx^2 )))/x)]](https://www.tinkutara.com/question/Q167886.png)
Commented by peter frank last updated on 28/Mar/22

Commented by MJS_new last updated on 29/Mar/22
![this is wrong. what you do in the 1^(st) and 3^(rd) lines makes no sense. t=((โ(1โx))/( (โ(1+x)))) โ x=((1โt^2 )/(1+t^2 )) dx=(dt/((d/dx)[((โ(1โx))/( (โ(1+x))))]))=(dt/(โ(1/( (โ(1โx))(โ((1+x)^3 ))))))=โ(โ(1โx))(โ((1+x)^3 ))dt โ โซ((โ(1โx))/(x(โ(1+x))))dx=โซ((โ(1โx))/(x(โ(1+x))))ร(โ(โ(1โx))(โ((1+x)^3 ))dt)= =โซ(((xโ1)(x+1))/x)dt=โซ((((โ2t^2 )/(t^2 +1))ร(2/(t^2 +1)))/(โ((t^2 โ1)/(t^2 +1))))dt= =4โซ(t^2 /((t^2 โ1)(t^2 +1)))dt](https://www.tinkutara.com/question/Q167913.png)
Commented by ArielVyny last updated on 29/Mar/22

Answered by MJS_new last updated on 28/Mar/22
![โซ((โ(1โx))/(x(โ(1+x))))dx= [t=((โ(1โx))/( (โ(1+x)))) โ dx=โ(โ(1โx))(โ((1+x)^3 ))dt] =4โซ(t^2 /(t^4 โ1))dt=โซ((2/(t^2 +1))+(1/(tโ1))โ(1/(t+1)))dt= =2arctan t +ln ((tโ1)/(t+1)) = =2arctan ((โ(1โx))/( (โ(1+x)))) +ln โฃ((1โ(โ(1โx^2 )))/x)โฃ +C](https://www.tinkutara.com/question/Q167895.png)