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f-is-a-function-increasing-and-C-1-on-a-b-prove-f-a-f-b-f-1-t-dt-a-b-x-f-x-dx-

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Question Number 29454 by prof Abdo imad last updated on 08/Feb/18
f is a function increasing and C^1 on [a,b] prove ∫_(f(a)) ^(f(b)) f^(−1) (t)dt = ∫_a ^b x f^′ (x)dx
fisafunctionincreasingandC1on[a,b]prove∫f(a)f(b)f−1(t)dt=∫abxf′(x)dx
Commented by prof Abdo imad last updated on 27/Feb/18
let put f^(−1) (t)=x ⇒t=f(x)⇒dt =f^′ (x)dx ∫_(f(a)) ^(f(b)) f^(−1) (t)dt= ∫_a ^(b ) x f^′ (x)dx .
letputf−1(t)=x⇒t=f(x)⇒dt=f′(x)dx∫f(a)f(b)f−1(t)dt=∫abxf′(x)dx.

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