If `f(a+b-x)=f(x),` then prove that `int_a^b xf(x)dx(a+b)/2int_a^bf(x)dxdot`

Explanation

We have to proof `int_a^b xf(x)dx=((a+b)/2)int_a^bf(x)dx`

Take , L.H.S

`I=int_a^b xf(x)dx....(i)`

As `int_a^b f(x)dx=int_a^b f(a+b-x)dx`

`I=int_a^b (a+b-x) f(a+b-x)dx`

`I=int_a^b (a+b-x) f(x)dx.....(ii)`

Now add (i) and (ii)

`2I=int_a^b f(x) [a+b-x+x]dx`

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