A) 1
B) 6
C) 2
D) 4
Correct Answer: A
Solution :
\[I=\int\limits_{2}^{4}{\frac{\log {{x}^{2}}}{\log {{x}^{2}}+\log \left( 6-x \right)}dx}\] ?.(1) Using property \[\int\limits_{a}^{b}{f\left( a+b-x \right)dx}=\int\limits_{a}^{b}{f\left( x \right)dx}\] \[I=\int\limits_{2}^{b}{\frac{\log {{\left( 6-x \right)}^{2}}}{\log {{\left( 6-x \right)}^{2}}+\log {{x}^{2}}}dx}\] ?(2) (1)+ (2) \[\Rightarrow \] \[I=\frac{1}{2}\int\limits_{2}^{4}{\frac{\cancel{\log {{x}^{2}}+\log {{\left( 6-x \right)}^{2}}}}{\cancel{\log {{x}^{2}}+\log {{\left( 6-x \right)}^{2}}}}}dx\] \[I=\frac{1}{2}\int\limits_{2}^{4}{dx}=\frac{1}{2}\left[ 4-2 \right]=1\]
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