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Question Number 125500 by mathmax by abdo last updated on 11/Dec/20
find relation between ∫ f(x)dx and ∫ f^(−1) (x)dx
findrelationbetween∫f(x)dxand∫f−1(x)dx
Answered by mathmax by abdo last updated on 11/Dec/20
let [a,b]⊂R^− we considere I=∫_a ^b f^(−1) (x)dx we do the changement f^(−1) (x)=t ⇒x=f(t)⇒I =∫_(f^(−1) (a)) ^(f^(−1) (b)) t f^′ (t)dt =_(by parts) [tf(t)]_(f^(−1) (a)) ^(f^(−1) (b)) −∫_(f^(−1) (a)) ^(f^(−1) (b)) f(t)dt =bf^(−1) (b)−af^(−1) (a)−∫_(f^(−1) (a)) ^(f^(−1) (b)) f(t)dt ⇒ ∫_a ^b f^(−1) (x) =bf^(−1) (b)−af^(−1) (a)−∫_(f^(−1) (a)) ^(f^(−1) (b)) f(x)dx
let[a,b]⊂R−weconsidereI=∫abf−1(x)dxwedothechangementf−1(x)=t⇒x=f(t)⇒I=∫f−1(a)f−1(b)tf′(t)dt=byparts[tf(t)]f−1(a)f−1(b)−∫f−1(a)f−1(b)f(t)dt=bf−1(b)−af−1(a)−∫f−1(a)f−1(b)f(t)dt⇒∫abf−1(x)=bf−1(b)−af−1(a)−∫f−1(a)f−1(b)f(x)dx

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