Definite Integration Question 349
Question: Let $ f:R\to R $ and $ g:R\to R $ be continuous functions, then the value of the integral $ \int _{-\pi /2}^{\pi /2}{[f(x)+f(-x)][g(x)-g(-x)]dx=} $
[IIT 1990; DCE 2000; MP PET 2001]
Options:
A) $ \pi $
B) 1
C) $ -1 $
D) 0
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ h(x)={f(x)+f(-x)}{g(x)-g(-x)} $
$ h(-x)={f(-x)+f(x)}{g(-x)-g(x)} $
$ =-{f(-x)+f(x)}{g(x)-g(-x)}=-h(x) $
Therefore, $ \int _{-\pi /2}^{\pi /2}{h(x)dx=0} $ .
BETA
