Consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = x\mathrm{e}^{x^2-3}$$
One of the antiderivatives $F$ of the function $f$ on $\mathbb{R}$ is defined by:\\
a. $F(x) = 2x\mathrm{e}^{x^2-3}$\\
b. $F(x) = \left(2x^2+1\right)\mathrm{e}^{x^2-3}$\\
c. $F(x) = \frac{1}{2}x\mathrm{e}^{x^2-3}$\\
d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2-3}$