The second fundamental theorem of Calculus says if a function is continuous (hence, Riemann integrable) then $\int_a ^x f(t) dt$ is a primitive of $f$. Ok this is for continuous functions.
However what happens for not necessarily continuous functions ? The first fundamental theorem of Calculus says if $f$ is Riemann integrable (i.e. not necessarily continuous) then $\int_a ^b f(t) dt = F(b) - F(a)$ where $F$ is a primitive.
Can $F$, in the context of the first fundamental theorem, be a primitive not of the form $\int_a ^x f(t) dt$ ?
Or do all the Riemann integrable functions have primitives of the form $\int_a ^x f(t) dt$ ?
Not all Riemann Integrable functions have primitives. Consider $f: [0,1]\to\mathbb{R}$ given by $f(x)=0, x\in[0,1/2)$ and $f(x)=1, x\in[1/2,1]$.
$f$ cannot be the derivative of a function because derivatives satisfy the Darboux condition and $f$ doesn't (it doesn't take on $1/2$, for example).
Now, $$f(x) = \cases{2x\sin\frac{1}{x^2}-\frac{2}{x}\cos\frac{1}{x^2} \quad\text{for } x\ne 0 \\0\quad\text{for }x=0}$$ is not continuous at $x=0$, but $$F(x) = \cases{x^2 \sin\frac{1}{x^2} \quad \text{for } x\ne 0 \\ 0\quad\text{for }x=0}$$
is a primitive of $f$.