Question

Solve ${\int }_2^4\frac{\log x^2}{\log x^2+\log (36-12x+x^2)}dx$

Answer 1

$I={\int }_2^4\frac{\log x^2}{\log x^2+\log {(x-6)}^2}$

$I={\int }_2^4\frac{\log {(6-x)}^{2}dx}{\log {(6-x)}^2+\log x^2}(\because {\int }_b^{a}f\left(x\right)dx={\int }_b^{a}f(a+b-x)dx)$

$\Rightarrow 2I={\int }_2^4\frac{\log x^2+\log {(6-x)}^2}{\log x^2+\log {(6-x)}^2}dx$

$\Rightarrow 2I={\int }_2^{4}1dx$

$2I=4-2$

$\Rightarrow I=1$Ans.