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"If "f: [-1,1]rarr[-(1)/(2),(1)/(2)]:f(x...

`"If "f: [-1,1]rarr[-(1)/(2),(1)/(2)]:f(x)=(x)/(1+x^(2)),` then find the area bounded by `y=f^(-1)(x),` the `x`-axis and the lines `x=(1)/(2), x=-(1)/(2).`

Text Solution

Verified by Experts

The correct Answer is:
`(1- log 2)` sq. units
`"Required area, "A=2overset(1//2)underset(0)intf^(-1)(x)dx`
`"Let "f^(1)(x)=t`
`rArr" "A=2overset(1)underset(0)inttf'(t)dt`
`=2[tf(t)]_(0)^(1)-2overset(1)underset(0)intf(t)dt`
`=2f(1)-overset(1)underset(0)int(2t)/(1+t^(2))dt`
`=1-[log (1+t^(2))]_(0)^(1)`
`=1-log 2`
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