Mathematics

If  $$f\left( {3 - x} \right) = f\left( x \right)$$  then  $$\int_1^2 {xf\left( x \right)} {\rm{ dx}}$$ equals-


ANSWER

$$\frac{3}{2}\int_1^2 {f\left( x \right)} {\rm{ dx}}$$


SOLUTION
$$f \left(3-x\right)=f\left(x\right)$$
Then, $$\int _{ 1 }^{ 2 }{ x } f\left( x \right) dx$$
$$=\int _{ 1 }^{ 2 }{ \left( 2+1-x \right)  } f\left( 3-x \right) dx$$
$$=\int _{ 1 }^{ 2 }{ \left( 3-x \right)  } f\left( x \right) dx$$
$$\Rightarrow \int _{ 1 }^{ 2 }{ xf\left( x \right)  } dx=\int _{ 1 }^{ 2 }{ 3f\left( x \right)  } -\int _{ 1 }^{ 2 }{ xf\left( x \right)  } dx$$
     $$2\int _{ 1 }^{ 2 }{ xf\left( x \right)  } dx=\int _{ 1 }^{ 2 }{ 3f\left( x \right)  } $$
     $$\int _{ 1 }^{ 2 }{ xf\left( x \right)  } dx=\dfrac { 3 }{ 2 } \int _{ 1 }^{ 2 }{ f\left( x \right)  } dx$$
Hence, the answer is $$\dfrac { 3 }{ 2 } \int _{ 1 }^{ 2 }{ f\left( x \right)  } dx.$$
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Single Correct Medium Published on 17th 09, 2020
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