Mathematics

If $$\displaystyle f\left ( a+b-x \right )= f\left ( x \right )$$ then $$\displaystyle \int_{a}^{b}xf\left ( x \right )dx$$ is equal to


ANSWER

$$\displaystyle \frac{a+b}{2}\int_{a}^{b}f\left ( x \right )dx$$


SOLUTION
As $$f\left( a+b-x \right) =f\left( x \right) $$
$$\displaystyle \int _{ a }^{ b }{ xf\left( x \right) dx } =\quad \displaystyle \int _{ a }^{ b }{ \left( a+b-x \right) f\left( a+b-x \right) dx } \\ =\displaystyle \int _{ a }^{ b }{ \left( a+b-x \right) f\left( x \right) dx } =\displaystyle \int _{ a }^{ b }{ \left( a+b \right) f\left( x \right) dx } -\displaystyle \int _{ a }^{ b }{ xf\left( x \right) dx } \\ \Rightarrow 2\displaystyle \int _{ a }^{ b }{ xf\left( x \right) dx } =\left( a+b \right) \displaystyle \int _{ a }^{ b }{ f\left( x \right) dx } \\ \Rightarrow \displaystyle \int _{ a }^{ b }{ xf\left( x \right) dx } =\quad \cfrac { \left( a+b \right)  }{ 2 } \displaystyle \int _{ a }^{ b }{ f\left( x \right) dx } $$
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Single Correct Medium Published on 17th 09, 2020
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