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If-f-x-is-a-differentiable-function-defined-x-R-such-that-f-x-3-x-f-x-0-then-0-2-f-1-x-dx-

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Question Number 115267 by bobhans last updated on 24/Sep/20
If f(x) is a differentiable function defined ∀x∈R such that (f(x))^3 −x+f(x)=0 then ∫_0 ^(√2) f^(−1) (x) dx =
Iff(x)isadifferentiablefunctiondefined∀x∈Rsuchthat(f(x))3−x+f(x)=0then∫20f−1(x)dx=
Answered by Olaf last updated on 24/Sep/20
f^3 (t)−t+f(t) = 0 Let t = f^(−1) (x) (fof^(−1) )^3 (x)−f^(−1) (x)+fof^(−1) (x) = 0 x^3 −f^(−1) (x)+x = 0 f^(−1) (x) = x^3 +x ∫_0 ^(√2) f^(−1) (x)dx = [(x^4 /4)+(x^2 /2)]_0 ^(√2) = 1+1 = 2
f3(t)−t+f(t)=0Lett=f−1(x)(fof−1)3(x)−f−1(x)+fof−1(x)=0x3−f−1(x)+x=0f−1(x)=x3+x∫02f−1(x)dx=[x44+x22]02=1+1=2

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