Definite Integration Question 52
Question: If $ f:R\to R $ and $ g:R\to R $ are one to one, real valued functions, then the value of the integral $ \int _{-\pi }^{\pi }{[f(x)+f(-x)][g(x)-g(-x)]dx} $ is
[DCE 2001; MP PET 2004]
Options:
A) 0
B) $ \frac{8}{3} $
C) 1
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ \varphi (x)=[f(x)+f(-x)][g(x)-g(-x)] $
then, $ \varphi (-x)=[f(-x)+f(x)][g(-x)-g(x)] $
$ \therefore \int _{-\pi }^{\pi }{\varphi (x)dx=0} $
therefore $ \int _{-\pi }^{\pi }{[f(x)+f(-x)][g(x)-g(-x)]dx=0} $ .
BETA
