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Let f be a one-to-one continuous functio...

Let `f` be a one-to-one continuous function such that `f(2)=3a n df(5)=7.G i v e nint_2^5f(x)dx=17 ,` then find the value of `int_3^7f^(-1)(x)dx`

Text Solution

Verified by Experts

The correct Answer is:
`12`
Let `y=f(x)`
`:.x=f^(-1)` and `dy=f'(x)dx`
`I=int_(3)^(7)f^(-1)(x)dx=int_(3)^(7)f^(-1)(y)dy`
Given `f^(-1)(3)=2` and `f^(-1)(7)=5`
`:. I=int_(2)^(5)xf'(x)dx=[x(f(x))]_(2)^(5)-int_(2)^(5)f(x)dx`
`=(5)(7)-(2)(3)-17=12`
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