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 a and b?

Expert Answers
justaguide eNotes educator| Certified Educator

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`

helga95 | Student

i'm sorry, but you are wrong again because my book got other response.  

my book got that the equation is true but must be shown how.please, don't give me wrong answer.