Indefinite Integral of a contant function

This question is bugging me

if an indefinite integral is of the form F(x) + C
stating that F(x) is an antiderivative of f(x) and since the derivative of a constant is 0 the collection of all of the anti derivatives are of the F(x) + C accounting for the fact any constant can be tagged to the end of that anti derivative..
now if this were an initial value problem you can take F(0) = 10 and solve for C correct...

well why with a constant function say f(x) = 1 when I solve for indefinite integral of the form F(x) + C .... I get x + C
well if f(x) is a constant function then wouldnt f(0) = 1 f(5) = 1
but in each of these cases i get C + 5 = 1 therefore c = -4 or C + 0 = 1 therefor C = 1

but if this if F(x) + C is a general expression that allows you to find the area under a graph up to point x ... then
find the area under a function which is constant f(x) = 1 on interval [0,9]
should = 9

but x + 1 = 9 + 1 = 10 so its obvious to see that C should equal zero

where am I going wrong can someone please clear this up?

$\int_{0}^{9}1dx = x + C|_{0}^{9} = (9 + C) - (0 + C) = 9 + C - C$