The value of the integral `int _(e ^(-1))^(e ^(2))|(ln x )/(x)|dx ` is:

Options

A. `(3)/(2)`
B. `(5)/(2)`
C. 3
D. 5

Answer

B

Explanation

We have,
`underset(e^(-1))overset(e^(2))int |(log_(e)x)/(x)|dx`
`rArr underset(e^(-1))overset(e^(2))int (|log_(e)x|)/(x)dx`
`rArrI= underset(1//e)overset(1)int (|log_(e)x|)/(x)dx+underset(1)overset(e^(2))int(|log_(e)x|)/(x)dx`
`rArrI=- underset(1//e)overset(1)int (log_(e)x)/(x)dx+underset(1)overset(e^(2))int(log_(e)x)/(x)dx`
`rArrI=- underset(1//e)overset(1)int log_(e)x d(log_(e)x) +underset(1)overset(e^(2))intlog_(e) x d(log_(e)x)`
`I=-[(1)/(2)(log_(e)x)^(2)]_(1//e)^(1)+[(1)/(2)(log_(e)x)^(2)]_(1)^(e^(2))`
`I=-[(1)/(2)xx0-(1)/(2)]+[(1)/(2)xx4-0]=(1)/(2)+2=(5)/(2)`