Definite Integration Question 39
Question: $ f(x)=f(2-x), $ then $ \int _{0.5}^{1.5}{xf(x)dx} $ equals
[AMU 1999]
Options:
A) $ \int _{0}^{1}{f(x)dx} $
B) $ \int _{0.5}^{1.5}{f(x)dx} $
C) $ 2\int _{0.5}^{1.5}{f(x)dx} $
D) 0
Show Answer
Answer:
Correct Answer: B
Solution:
$ I=\int _{0.5}^{1.5}{xf(x)dx=\int _{0.5}^{1.5}{(2-x)f(2-x)dx}} $ , $ [ \because \int_a^{b}{f(x)dx=\int_a^{b}{f(a+b-x)dx}} ] $
$ =\int _{0.5}^{1.5}{(2-x)f(x)dx}=2\int _{0.5}^{1.5}{f(x)dx-I} $
therefore $ I=\int _{0.5}^{1.5}{f(x)dx} $ .
BETA
