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

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

Text Solution

Verified by Experts

The correct Answer is:
`"log"|(xe^(x))/(1+xe^(x))|+(1)/(1+xe^(x))+c`
Let `I=int((x+1))/(x(1+xe^(x))^(2))dx=int(e^(x)(x+1))/(xe^(x)(1+xe^(x))^(2))dx`
Put `1 + xe^(x) = t rArr (e^(x)+xe^(x))dx=dt`
`therefore" "I=int(dt)/((t-1)t^(2))=int[(1)/(t-1)-(1)/(t)-(1)/(t^(2))]dt`
`=log|t-1|-log|t|+(1)/(t)+c`
`=log|(t-1)/(t)|+(1)/(t)+c`
`=log|(xe^(x))/(1+xe^(x))|+(1)/(1+xe^(x))+c`
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