Is it true that the definite integral of x*f(x) = ((a+b)/2)*definite integral of f(x), where f(x) = a + b - x and the limits of the integration are...

It is given that f(x) = a + b -
x

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

= `int_a^b ax + bx - x^2
dx`

= a(b^2 - a^2)/2 + b(b^2 - a^2)/2 - (b^3 -
a^3)/3

= ab^2/2 - a^3/2 + b^3/2 - ba^2/2 - b^3/3 + a^3/3
...(1)

[(a + b)/2]`int_a^bf(x)dx` = [(a + b)/2]*`int_a^b(a
+ b - x)dx`

= [(a + b)/2]*[a(b - a) + b(b - a) - (b^2 -
a^2)/2]

=   a(b^2 - a^2)/2 + b(b^2 - a^2)/2 -  (b^3 + a*b^2
- a^2*b - a^3)/4

= ab^2/2 - a^3/2 + b^3/2 - ba^2/2 - b^3/4
- ab^2/4 - a^2*b/4 - a^3/4 ...(2)

(1) and (2) are not
equal.

It is not true that `int_a^b x*f(x)dx` = [(a +
b)/2]*`int_a^bf(x)dx`