Let `f` be a real-valued function satisfying `f(x)+f(x+4)=f(x+2)+f(x+6)` Prove that `int_x^(x+8)f(t)dt` is constant function.

Explanation

Given that `f(x)+f(x+4)=f(x+2)+f(x+6)`…………1
Replacing `x` by `x+2` we get
`f(x+2)+f(x+6)=f(x+4)+f(x+8)`…………………2
From equation 1 and 2 we get `f(x)=f(x+8)`……………3
or `int_(x)^(x+8)f(t)dt+int_(0)^(8)f(t)dt`
Thus, `g` is a constant function.