Mathematics

# If $\displaystyle f\left ( \frac{1}{x} \right )+x^{2}f\left ( x \right )=0$ for $x> 0,$ and $\displaystyle I=\int_{1/x}^{x}f\left ( z \right )dz, \frac{1}{2}\leq x\leq 2$ then $\displaystyle I$ is?

$0$

##### SOLUTION
Substitute $\displaystyle z=\dfrac { 1 }{ t }$

$\displaystyle I=\int _{ x }^{ \dfrac { 1 }{ t } }{ f\left( \dfrac { 1 }{ t } \right) } .\left( \dfrac { -1 }{ { t }^{ 2 } } \right) dt=\int _{ x }^{ \dfrac { 1 }{ x } }{ \left( -{ t }^{ 2 }.f\left( t \right) \right) } \left( \dfrac { -1 }{ { t }^{ 2 } } \right)$

$\displaystyle =\int _{ x }^{ \dfrac { 1 }{ x } }{ f\left( t \right) } dt=-\int _{ \dfrac { 1 }{ x } }^{ x }{ f\left( t \right) } dt=-I\\ \therefore 2I=0$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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