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The integral int(1)^(e){((x)/(e))^(2x)-...

The integral `int_(1)^(e){((x)/(e))^(2x)-((e)/(x))^(x)} "log"_(e)x` dx is equal to

A

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

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
B
Put `((x)/(e))^(x) = t " "x rarr 1, t rarr e^(-1)`
`x rarr e, t rarr 1, x^(x).e^(x) = t`
`[x^(x)[ln x + 1] e^(-x) - e^(-x)x^(x)] dx = dt`
`x^(x) = lnx.e^(-x) dx = dt`
`[ln xdx][x^(x).e^(-x)] = dt , ln x.dx = (dx)/(t)`
`int_(e^(-1))^(1)[t^(2) -(1)/(t)](dt)/(t)rArr int_(e^(-1))^(1)[t - (1)/(t^(2))]dt = [(t^(2))/(2)+(1)/(t)]_(-e^(-1))^(1) = (1)/(2)+1 - (e^(-2))/(2) - e = (3)/(2) - e - (1)/(2e^(2))`
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