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The integral int(-1/2)^(1/2) ([x]+1n((1+...

The integral `int_(-1/2)^(1/2)` `([x]+1n((1+x)/(1-x)))dx` is equal to (where [.] represents the greatest integer function) `-1/2` (b) 0 (c) 1 (c) `21n(1/2)`

A

`-(1)/(2)`

B

0

C

1

D

`"log"((1)/(2))`

Text Solution

Verified by Experts

The correct Answer is:
A
`int_(-1//2)^(1//2)[[x]+log((1+x)/(1-x))]dx`
`=int_(-1//2)^(1//2)[x]dx+int_(-1//2)^(1//2)log((1+x)/(1-x))`dx
`=int_(-1//2)^(1//2)[x]dx+0[:' log((1+x)/(1-x)) " is an odd function "]`
`=int_(-1//2)^(0)[x]dx+int_(0)^(1//2)[x]dx=int_(-1//2)^(0)(-1)dx+int_(0)^(1//2)(0)dx`
`=-[x]_(-1//2)^(0)=-(0+(1)/(2))=-(1)/(2)`
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