Antiderivative of the Jacobian of a vector function on a line

Question

$\newcommand{\R}{\mathbb R}$ Let $f\colon \R^m\to\R^n$ be differentiable and $a,b\in\R^m$. Denote the Jacobian of $f$ as $g\colon \R^m \to \R^{n\times m}$. Consider the integral $$\int_0^1 g(a+\theta b)\,d\theta.$$ Is there a way to construct an antiderivative $G\colon [0,1]\to \R^{n\times m}$ of the function $\theta\mapsto g(a+\theta b)$ such that $$\int_0^1 g(a+\theta b)\,d\theta = G(1) - G(0)?$$ Can $G$ be expressed in terms of $f$?