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Given a real-valued function f which is ...

Given a real-valued function f which is monotonic and differentiable. Then `int_(f(a))^(f(b))2x(b-f^(-1)(x))dx=`

A

`int_(a)^(b)(f^(2)(x)-2f^(2)(a))dx`

B

`int_(a)^(b)(2f^(2)(x)-f^(2)(a))dx`

C

`int_(a)^(b)(f^(2)(x)-f^(2)(a))dx`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Let `f^(-1)(x)=u`
`therefore" "int_(f(a))^(f(b))2x(b-f^(-1)(x))dx`
`" "=int_(1)^(b)2f(u)(b-u)f'(u)du`
`" "=|(b-u)f^(2)(u)|_(a)^(b)+int_(a)^(b)f^(2)(u)du`
`" "=int_(a)^(b)(f^(2)(x)-f^(2)(a))dx`
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