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int(1//3)^(3)(1)/(x)log(e)(|(x+x^(2)-1)/...

`int_(1//3)^(3)(1)/(x)log_(e)(|(x+x^(2)-1)/(x-x^(2)+1)|)dx` is equal to

A

`(8)/(3)`

B

`-(8)/(3)`

C

0

D

3

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int_(1//3)^(2)(1)/(x)log_(e)(|(x+x^(2)-1)/(x-x^(2)+1)|)dx`
Let `x=(1)/(t)rArr dx =-(1)/(t^(2))dt`
`rArr" "I=-int_(3)^(1//3)tlog_(e)(|((1)/(t)+(1)/(t^(2))-1)/((1)/(t)-(1)/(t^(2))+1)|)(1)/(t^(2))dt`
`" "=int_(1//3)^(3)(1)/(t)log_(e)(|(t-t^(2)+1)/(t+t^(2)-1)|)dt`
`" "=-int_(1//3)^(3)(1)/(x)log_(2)(|(x+x^(2)-1)/(x-x^(2)+1)|)dx`
`rArr" "I=-IrArr I=0`
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