Let $g:[1,\infty )\to [2,\infty )$  be a differentiable function satisfying $6\underset{1}{\overset{x}{\int }}g\left(t\right)dt=3xg\left(x\right)-x^3-5\forall x\ge 1$  then $g\left(2\right)=$   ……. 
 

$$\begin{gathered} Put x=1\Rightarrow g\left(1\right)=2 \\ Diff. 6g\left(x\right)=3\left(x{g}^1\left(x\right)+g\left(x\right)\right)-3x^2 \\ \frac{x{g}^1\left(x\right)-g\left(x\right)}{x^2}=1 \\ \frac{d}{dx}\left(\frac{g\left(x\right)}{x}\right)=1 \\ \frac{g\left(x\right)}{x}=x+c\Rightarrow g\left(x\right)=x(x+1) \\ g\left(2\right)=2\left(3\right)=6 \end{gathered}$$