If-f-x-f-a-b-x-then-a-b-x-f-x-dx-

Question Number 54034 by qw last updated on 28/Jan/19

If f (x) = f (a + b - x), then \underset{a}{\int^{b}} x f (x) d x =

Commented by maxmathsup by imad last updated on 28/Jan/19

\int_{a}^{b} x f (x) d x = \int_{a}^{b} x f (a + b - x) d x c h a n g e m e n t a + b - x = t g i v e

\int_{a}^{b} x f (x) d x = \int_{a}^{b} (a + b - t) f (t) d t = \int_{a}^{b} (a + b) f (t) - \int_{a}^{b} t f (t) d t \Rightarrow

2 \int_{a}^{b} x f (x) = (a + b) \int_{a}^{b} f (x) \Rightarrow \int_{a}^{b} x f (x) d x = \frac{a + b}{2} \int_{a}^{b} f (x) d x .