Mathematics - Definite Integration Question with Solution | TestHub

Let $I={\int }_5^9\frac{\log 3x^{2}dx}{\log 3x^2+\log \left(588-84x+3x^2\right)}$ ...(i) $\begin{array}{rl}& ={\int }_5^9\frac{\log 3x^{2}dx}{\log 3x^2+\log 3(14-x{)}^2}\\ & ={\int }_5^9\frac{\log 3(14-x{)}^{2}dx}{\log 3(14-x{)}^2+\log 3(14-(14-x){)}^2}\\ & \left[{\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx\right]\end{array}$ $I={\int }_5^9\frac{\log 3(14-x{)}^{2}dx}{\log 3(14-x{)}^2+\log 3x^2}$ ...(ii) Adding Eqs. (i) and (ii), we get $\begin{array}{rl}& 2I={\int }_5^9\frac{\log 3x^2+\log 3(14-x{)}^2}{\log 3(14-x{)}^2+\log 3x^2}dx\\ & 2\mathrm{I}={\int }_5^{9}dx=9-5=4\Rightarrow I=2\end{array}$