A-B-R-f-1-0-0-1-f-x-2-dx-A-and-0-1-xf-x-dx-B-what-is-the-integral-value-of-0-1-xf-x-f-x-1-dx-by-using-trrms-of-A-and-B-

Question Number 138415 by tugu last updated on 13/Apr/21

A, B \in R, f (1) = 0, \underset{0}{\int^{1}} {(f (x))}^{2} d x = A a n d \underset{0}{\int^{1}} x f (x) d x = B

w h a t i s t h e i n t e g r a l v a l u e o f \underset{0}{\int^{1}} x f (x) (f^{'} (x) - 1) d x b y u s i n g t r r m s o f A a n d B ?

Answered by Ar Brandon last updated on 13/Apr/21

I = \int_{0}^{1} x f (x) (f^{'} (x) - 1) d x

= \int_{0}^{1} x f (x) f^{'} (x) d x - \int_{0}^{1} x f (x) d x

= {[\frac{x {(f (x))}^{2}}{2} - \frac{1}{2} \int {(f (x))}^{2} d x]}_{0}^{1} - B

= \frac{{(f (1))}^{2}}{2} - \frac{1}{2} \int_{0}^{1} {(f (x))}^{2} d x - B

= - \frac{A}{2} - B