已知函数 $f(x)=\left\{\begin{array}{cc}2(x-1), & x < 1, \\ \ln x, & x \geqslant 1,\end{array}\right.$ 则 $f(x)$ 的一个原函数是
$\text{A.}$ $F(x)=\left\{\begin{aligned}(x-1)^2, & x < 1, \\ x(\ln x-1), & x \geqslant 1 .\end{aligned}\right.$
$\text{B.}$ $F(x)=\left\{\begin{array}{cc}(x-1)^2, & x < 1, \\ x(\ln x+1)-1, & x \geqslant 1 .\end{array}\right.$
$\text{C.}$ $F(x)=\left\{\begin{array}{cc}(x-1)^2, & x < 1, \\ x(\ln x+1)+1, & x \geqslant 1 .\end{array}\right.$
$\text{D.}$ $F(x)=\left\{\begin{array}{cc}(x-1)^2, & x < 1, \\ x(\ln x-1)+1, & x \geqslant 1 .\end{array}\right.$
$\text{E.}$
$\text{F.}$