I'm an idiot

The integral of f(x) over [a,b] is equal to the integral of f(x) over [a,c] plus the integral of f(x) over [c,b]

Note that f(x) must be the same over this interval. Silly me. :D

(My only consolation is that not only did my two classmates fail to spot this, but the student teacher was the one who tried to do this with different functions and led us into the incorrect solution. :r)