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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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