Definite Integration Question 482
Question: If $ f(a+b-x)=f(x), $ then $ \int_a^{b}{xf(x)dx=} $
[CEE 1993; AIEEE 2003]
Options:
A) $ \frac{a+b}{2}\int_a^{b}{f(b-x)dx} $
B) $ \frac{a+b}{2}\int_a^{b}{f(x)dx} $
C) $ \frac{b-a}{2}\int_a^{b}{f(x)dx} $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Since $ I=\int_a^{b}{xf(x)dx=\int_a^{b}{(a+b-x)f(a+b-x)dx}} $
therefore $ I=\int_a^{b}{(a+b)}f(x)dx-\int_a^{b}{xf(x)dx} $
$ { \because f(a+b-x)=f(x)given } $
therefore $ 2I=(a+b)\int_a^{b}{f(x)dx} $
therefore $ I=\int_a^{b}{xf(x)dx=\frac{a+b}{2}\int_a^{b}{f(x)dx}} $ .
BETA
