TITLE: Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$.
QUESTION [21 upvotes]: This question was asked in an entrance test for an undergraduate mathematics program today, held all over India. 
Question: $f$ is a differentiable function in $[0,1]$ such that $f(f(x))=x$ and $f(0)=1$.
Find the value of $\int_{0}^{1} (x-f(x))^{2016} dx$.
I tried to solve it by substituting $f(f(x))$ in place of $x$, but could not proceed much further. Any hints or solutions will be appreciated.

REPLY [3 votes]: An alternative solution is to note that $f(f(x))=x$ means that $f(x)$ is its own inverse. Geometrically, this means that the function will be perpendicular to the line $y=x$ at the point of intersection and symmetric on each side of $y=x$. Since it is differentiable over $[0,1]$ and we are given that $f(0)=1$, one such function that comes to mind is $f(x)=1-x$. 
Now that a possible function is known, the calculation of the integral is easy:
$$\int_0^1 (x-f(x))^{2016}\,dx=\int_0^1(2x-1)^{2016}\,dx=\frac{1}{2017}$$