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

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

Text Solution

Verified by Experts

`I=int(logx)/((1+logx)^(2))dx`
Let `logx=t. " Then " x=e^(x)`
or `dx=e^(t) dt.` Thus,
`I=int(te^(t))/((t+1)^(2))dt`
`=int e^(t)((1)/((t+1))-(1)/((t+1)^(2)))dt`
`=(e^(t))/(t+1)+c`
`=(x)/((log x+1))+c`
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