Chọn phương án A.

\(\begin{aligned}
F\left(x\right)&=\displaystyle\int f\left(x\right)\mathrm{\,d}x=\displaystyle\int\dfrac{x+2}{x-1}\mathrm{\,d}x\\
&=\displaystyle\int\dfrac{x-1+3}{x-1}\mathrm{\,d}x\\
&=\displaystyle\int\left(1+\dfrac{3}{x-1}\right)\mathrm{\,d}x\\
&=x+3\ln\left|x-1\right|+C.
\end{aligned}\)

Vì \(x\in(1;+\infty)\) nên \(F\left(x\right)=x+3\ln\left(x-1\right)+C\).