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The integral underset1overseteint{(x/e)^...

The integral `underset1overseteint{(x/e)^(2x)-(e/x)^x}log_exdx` is equal to

A

`(3)/(2)-e-(1)/(2e^(2))`

B

`-(1)/(2)+(1)/(e)-(1)/(2e^(2))`

C

`(1)/(2)-e-(1)/(e^(2))`

D

`(3)/(2)-(1)/(e)-(1)/(2e^(2))`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `I=int_(1)^(e){((x)/(e))^(2x)-((e)/x)^(x)} "log"_(e) x dx`
Now , put `((x)/(e))^(x)=t rArrx log _(e)((x)/(e))= log t`
`rArrx (log_(e)x-log_(e))=logt`
`rArr[x((1)/(x))+(log_(e)x-log_(e)e)]dx=(1)/(t)dt`
`rArr(1+log_(e)x-1)dx=(1)/(t)dtrArr(log_(e)x)dx=(1)/(t)dt`
Also , upper limit `x=erArrt=1` and lower limit `x=1rArrt=(1)/(e)`
`I=int_(1//e)^(1)(t^(2)-(1)/(t))*(1)/(t)dtrArrI=int_(1//e)^(1)(t-t^(2)) dt`
`I=[((t^(2))/(2)+(1)/(t))]_(1/(e))^(1)={((1)/(2)+1)-((1)/(2e^(2))+e)}=(3)/(2)-e-(1)/(2e^(2))`
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