Let, $I={\int }_5^9\frac{{\log }_{\mathrm{e}}3{\mathrm{x}}^2}{{\log }_{\mathrm{e}}3{\mathrm{x}}^2+{\log }_{\mathrm{e}}\left(588-84\mathrm{x}+3{\mathrm{x}}^2\right)}\mathrm{dx}$

$\Rightarrow I={\int }_5^9\frac{{\log }_{\mathrm{e}}3{\mathrm{x}}^2}{{\log }_{\mathrm{e}}3{\mathrm{x}}^2+{\log }_{\mathrm{e}}3\left(196-28\mathrm{x}+{\mathrm{x}}^2\right)}\mathrm{dx}$

$\Rightarrow I={\int }_5^9\frac{{\log }_{\mathrm{e}}3{\mathrm{x}}^2}{{\log }_{\mathrm{e}}3{\mathrm{x}}^2+{\log }_{\mathrm{e}}3{\left(14-\mathrm{x}\right)}^2}\mathrm{dx} ....\left(1\right)$

$\Rightarrow I={\int }_5^9\frac{{\log }_{\mathrm{e}}3{\left(9+5-\mathrm{x}\right)}^2}{{\log }_{\mathrm{e}}3{\left(9+5-\mathrm{x}\right)}^2+{\log }_{\mathrm{e}}3{\left(14-\left(9+5-\mathrm{x}\right)\right)}^2}\mathrm{dx}$

$\Rightarrow I={\int }_5^9\frac{{\log }_{\mathrm{e}}3{\left(14-\mathrm{x}\right)}^2}{{\log }_{\mathrm{e}}3{\left(14-\mathrm{x}\right)}^2+{\log }_{\mathrm{e}}3{\mathrm{x}}^2}\mathrm{dx} .....\left(2\right)$

Adding $\left(1\right) \& \left(2\right)$, we have

$2\mathrm{I}={\int }_5^9\frac{{\log }_{\mathrm{e}}3{\mathrm{x}}^2+{\log }_{\mathrm{e}}3{\left(14-\mathrm{x}\right)}^2}{{\log }_{\mathrm{e}}3{\mathrm{x}}^2+{\log }_{\mathrm{e}}3{\left(14-\mathrm{x}\right)}^2}d\mathrm{x}$

$$\begin{gathered} \Rightarrow 2\mathrm{I}={\int }_5^9\mathrm{dx} \\ \Rightarrow 2\mathrm{I}={\left[\mathrm{x}\right]}_5^9 \\ \Rightarrow 2\mathrm{I}=9-5 \\ \Rightarrow \mathrm{I}=2 \end{gathered}$$