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The value of the integral int(e^(-1))^(e...

The value of the integral `int_(e^(-1))^(e^2)|((log)_e x)/x|dxi s` `3/2` (b) `5/2` (c) 3 (d) 5

A

`3//2`

B

`5//2`

C

3

D

5

Text Solution

Verified by Experts

The correct Answer is:
D
`int_(e^(-1))^(e^(2))|(log_(e)x)/(x)|dx=int_(e^(-1))^(1)|(log_(e)x)/(x)|dx-int_(1)^(e^(2))|(log_(e)x)/(x)|dx`
`["since , 1 is turning point for"|(log_(e)x)/(x)|" for + ve and - ve values"]`
`=-int_(e^(-1))^(1)(log_(e)x)/(x)dx+int_(1)^(e^(2))|(log_(e)x)/(x)|dx`
`=-(1)/(2)[(log_(e)x)^(2)]_(e^(-1))^(1)+(1)/(2)[(log_(e)x)^(2)]_(1)^(e^(2))`
`=-(1)/(2){0-(-1)^(2)}+(1)/(2)(2^(2)-0)=(5)/(2)`
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