Mathematics

$$\displaystyle \int _{ 0 }^{ a }{ f\left( x \right) +f\left( -x \right)  } dx$$ is equal to


ANSWER

$$\displaystyle \int_{-a}^{a}f (x)dx$$


SOLUTION
Using property $$\int _{ a }^{ b }{ f\left( x \right) dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx } $$, 

we get
$$\int _{ -a }^{ a }{ f\left( x \right) dx } =\int _{ 0 }^{ a }{ f\left( x \right) dx } +\int _{ 0 }^{ a }{ f\left( -a+a-x \right) dx } =\int _{ 0 }^{ a }{ f\left( x \right) dx } +\int _{ 0 }^{ a }{ f\left( -x \right) dx }  =\int _{ 0 }^{ a }{ \left( f\left( x \right) +f\left( -x \right)  \right) dx } $$

$$\int _{ 0 }^{ a }{ \left( f\left( x \right) +f\left( -x \right)  \right) dx } =\int _{ -a }^{ a }{ f\left( x \right) dx } $$
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Single Correct Medium Published on 17th 09, 2020
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