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int(1+x)/(x+e^(-x))dx is equal to...

`int(1+x)/(x+e^(-x))dx` is equal to

A

`log|(x-e^(-x))|+C`

B

`log|(x+e^(-x))|+C`

C

`log|(1+xe^(x))|+C`

D

`(1+xe^(x))^(2)+C`

Text Solution

Verified by Experts

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