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Evaluate: int(1)/(x)ln((x)/(e^(x)))dx=...

Evaluate: `int(1)/(x)ln((x)/(e^(x)))dx=`

A

`(1)/(2)e^(x)-lnx+C`

B

`(1)/(2)lnx-e^(x)+C`

C

`(1)/(2)(lnx)^(2)-x+C`

D

`(e^(x))/(2x)+C`

Text Solution

Verified by Experts

The correct Answer is:
C
`int(1)/(x)ln((x)/(e^(x)))dx`
`=int(1)/(2)(lnx-lne^(x))dx`
`=int(ln x-x)/(x)dx`
`=[int(1)/(x)lnxdx-int (1)/(x)xdx]`
`=(1)/(2)(ln x)^(2)-x+C`
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