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 =

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