If I have a certain $$F(x)=\int_{a(x)}^{b(x)} f(t)dt$$
I know that $F'(x)=f(b(x))\cdot b'(x) + f(a(x))\cdot a'(x)$.
Now, my question is that: can I write $F$ as an integral function? So something like $$F(x)=\int_{p}^{x} F'(t)dt=\int_{p}^{x}f(b(t))\cdot b'(t) + f(a(t))\cdot a'(t)dt$$with $p\in \mathbb{R}$.
If it's possible, how do I find $p$?