Let's first confine ourselves to a continuous function $f:\mathbb{R}\to\mathbb{R}$ over an interval $[a,b]$. When we define $$\int_a^bf(x) dx$$ we usually consider the Darboux integral or the Riemann Integral. And, further using these definitions, we prove the properties of definite integrals. And then, there is the second FTOC, which states that if $F$ is anti-derivative of $f$, then $$\int_a^bf(x)dx=F(b)-F(a)$$ I was wondering, for a definition, why we don't use this. I don't really know, but if this theorem works for all functions, whether continuous or not, which have a definite integral, then using this theorem as the definition would really make the proofs of properties of integrals easier.
So, my question: Why don't we use this as a definition?