On line 4. The second expression wouldn't really include an (indefinite integral) integration constant. You evaluate at "x" and "0" and get
F(x) - F(0)
which should be the third expression. There is no arbitrary "C" in this expression. The derivative of this
f(x)
represents how the "area" changes as you vary the upper limit "x". It does not depend on the lower limit F(0) as i) its a constant and ii) you're interested in the change in area, so this does not depend on any starting point.
The 4th line effectively says an indefinite integral is equal to a definite integral which is not correct.