\(\int_{2}^{4}\frac{\log x^2}{\log x^2+\log(36-12x+x^2)}dx=\)
Answer & explanation
Correct answer: option 1
\(I=\int\limits_{2}^{4}\frac{\log x^2}{\log x^2+\log(36-12x+x^2)}dx=\int\limits_{2}^{4}\frac{\log x}{\log x+\log (x-6)}dx\) ...(1)
$I=\int\limits_{2}^{4}\frac{\log (6-x)}{\log x+\log (6-x)}dx$ ...(2) [Using $\int\limits_{a}^{b}f(x)dx=f(a+b-x)dx$]
Eq. (1) + Eq. (2)
$2I=\int\limits_{2}^{4}1dx⇒I=1$