Mathematics

Solve $$\displaystyle\int^{4}_{2}\dfrac {\log x^{2}}{\log x^{2}+\log (36-12x+x^{2})}dx$$


ANSWER

1


SOLUTION
$$ I=\displaystyle \int _{ 2 }^{ 4 }{ \dfrac { \log { { x }^{ 2 } }  }{ \log { { x }^{ 2 } } +\log { { \left( x-6 \right)  }^{ 2 } }  }  }$$
$$ I=\displaystyle \int _{ 2 }^{ 4 }{\dfrac{\log { { \left( 6-x \right)  }^{ 2 } }dx}{\log { { \left( 6-x \right)  }^{ 2 } }+\log{{x}^{2}}}} \left(\because \int_{b}^{a}{f\left(x\right)dx}=\int_{b}^{a}{f\left(a+b-x\right)dx} \right)$$
$$\Rightarrow 2I=\displaystyle \int _{ 2 }^{ 4 }{ \dfrac { \log { { x }^{ 2 } } +\log { { \left( 6-x \right)  }^{ 2 } }  }{ \log { { x }^{ 2 } } +\log { { \left( 6-x \right)  }^{ 2 } }  }  }dx$$
$$\Rightarrow 2I=\displaystyle \int_{2}^{4}{1}dx$$
$$2I=4-2$$
$$\Rightarrow \boxed{I=1}$$Ans.
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