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Evaluate: int(logx)/((1+logx)^2)dx...

Evaluate: `int(logx)/((1+logx)^2)dx`

Text Solution

Verified by Experts

`"Let I"=int (log x)/((1+logx)^(2))dx`
`"Put log "x=t," Then "x=e^(t)" "therefore dx=x^(t)dt`
`therefore I=int (t)/((1+t)^(2))e^(t) dt=int e^(t) [((1+t)-1)/((1+t)^(2))]dt`
`=int e^(t) [(1)/(1+t)-(1)/((1+t)^(2))]dt`
`"If "f(t)=(1)/(1+t)," then "f'(t)=(-1)/((1+t)^(2))`
`therefore I=int e^(t)[f(t)+f'(t)]dt=e^(t)f(t)+c=(e^(t))/(1+t)+c=(x)/(1+logx)+c`
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