Chọn phương án C.

Đặt $\begin{cases}
u=2x-5\\
v'=f'(x)
\end{cases}\Rightarrow\begin{cases}
u'=2\\
v=f(x).
\end{cases}$. Khi đó $$\begin{aligned}[t]
\displaystyle\int\limits_1^6(2x-5)f'(x)\mathrm{~d}x&=(2x-5)f(x)\bigg|_1^6-2\displaystyle\int\limits_1^6f(x)\mathrm{~d}x\\
&=\big[7f(6)+3f(1)\big]-2\left[\dfrac{(3+8)\cdot5}{2}-\dfrac{125}{9}\right]\\
&=7\cdot8+3\cdot3-2\cdot\dfrac{245}{18}=\dfrac{340}{9}.
\end{aligned}$$