Mathematics

# Let $f:R\rightarrow R$ and $g:R\rightarrow R$  be continuous functions, then the value of $\displaystyle\int_{\displaystyle-\frac{\pi}{2}}^{\displaystyle\frac{\pi}{2}}{(f(x)+f(-x))(g(x)-g(-x))dx}$, is equal to

##### ANSWER

$0$

##### SOLUTION
Let ${ I }=\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( f\left( x \right) +f\left( -x \right) \right) \left( g\left( x \right) -g\left( -x \right) \right) dx }$ ...(1)
Using property $\int _{ a }^{ b }{ f\left( x \right) dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx }$
${ I }=\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( f\left( -x \right) +f\left( x \right) \right) \left( g\left( -x \right) -g\left( x \right) \right) dx }$ ...(2)
Adding (1) and (2), we get
$2I=\int _{ -\cfrac { \pi }{ 2 } }^{ \cfrac { \pi }{ 2 } }{ \left( f\left( -x \right) +f\left( x \right) \right) \left( \left( g\left( x \right) -g\left( -x \right) \right) +\left( g\left( -x \right) -g\left( x \right) \right) \right) dx } \\ =0\Rightarrow I=0$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
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