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

Options

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))`

Answer

B

Explanation

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))`