2015_32 Simple and elegant Part 3

Calculus Level pending

Let $$f\left( x \right)$$ be a real valued function not identically zero such that $f\left( x+{ y }^{ n } \right) =f\left( x \right) +{ (f(y)) }^{ n }\quad \forall x,y\epsilon R$ where $$n\epsilon N(n\neq 1)$$ and $$f^{ ' }\left( 0 \right) \ge 0$$ we may get an explixity form of the function $$f\left( x \right)$$.

Then the value of $$\int _{ 0 }^{ 1 }{ f\left( x \right) dx }$$ is?