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Integrate the following w.r.t.x.: (log...

Integrate the following w.r.t.x.:
`(log(x+2)-logx)/(x(x+2))`

Text Solution

Verified by Experts

`"Let I"=int(log(x+2)-logx)/(x(x+2))dx`
`=int[log(x+2)-logx].(dx)/(x(x+2))`
Put t=log(x+2)-logx. Then
`dt=((1)/(x+2)-(1)/(x))dx=(x-x-2)/(x(x+2))dx=(-2)/(x(x+2))dx`
`therefore (dx)/(x(x+2))=-(dt)/(2)`
`therefore I=int t((-dt)/(2))=-(1)/(2) int dt=-(1)/(2).(t^(2))/(2)+c=-(1)/(4)[log(x+2)-logx]^(2)+c`
`=-(1)/(4)[log ((x+2)/(x))]^(2)+c`
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