Love for math 1

SO LET'S START AS WE CAN SEE THAT ITS GIVEN THAT F(F(X))=X WHICH BRINGS US TO THE CONCLUSION THAT F(X) CAN BE ANY INVERSE TRIGONOMETRIC FUNCTION OR F(X) WOULD BE OF THE FORM F(X)=C-X BUT SINCE ANOTHER CONDITION IS GIVEN THAT F(0)=1 F(X) CAN'T BE ANY INVERSE TRIGONOMETRIC FUNCTION WHICH LEAVES US WITH F(X)=C-X AT X=0 F(0)=1-X SO OUR FUNCTION IS F(X)=1-X NOW SOLVING THE INTEGRAL I= $\int\limits_{0}^{1}{[X-F(X)]^{2016} }dx$

ON PUTTING F(X)=1-X WE GET I= $\int\limits_{0}^{1}{[2X-1]^{2016} }dx$ NOW USING THE PROPERTY OF INTEGRATION = $[\frac{(ax+b)^{n+1} }{a\times (n+1)} ]^{s}_{d}$ = $[\frac{(2X-1)^{2017} }{2\times 2017} ]^{1}_{0}$ ON PUTTING THE LIMITS, = $\frac{1}{4034} + \frac{1}{4034}$ = $\frac{2}{4034}$ = $\frac{1}{2017}$ HOPE IT HELPS!!!!