Mathematics

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

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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One Word Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
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