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Evaluate : ∫(1+logx)/(x(2+logx)(3+logx))dx - Mathematics and Statistics

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Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`

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Solution

Let `I=int (1+logx)/(x(2+logx)(3+logx))dx`

Put 

`logx=t`

`1/xdx=dt`

`I=int(1+t)/((2+t)(3+t))dt`

consider 

`(1+t)/((2+t)(3+t))=A/(2+t)+B/(3+t)`

`(1+t)=A(3+t)+B(2+t)`

A=-1,B=2

`(1+t)/((2+t)(3+t))=-1/(2+t)+2/(3+t)`

`I=int-1/(2+t)dt+int2/(3+t)dt`

`=-log|(2+t)|+2log|(3+t)|+c`

`=log[|((3+t)^2)/(2+t)|] +c`

`=log[|(3+logx)^2/(2+logx)|]+C`

Concept: Indefinite Integration - Methods of Integration
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