If $f$ is a continuous, real-valued function on interval $[a,b]$, then the fundamental theorem of calculus tells us that
where $F(x)$ is antiderivative, i.e. $F(x)'=f(x)$.
If so, why I can't find the equality $\int f(x)dx=\int_a^x f(t)dt=F(x)$ anywhere? It expresses the relationship between definite and indefinite integral in such a straightfoward way (assuming this equality is true). So it it true and can I use $\int$ and $\int_a^x$ interchangeably?