Integral Calculus Question 203
Question: Let $ f:R\to R $ and $ g:R\to R $ be continuous functions. Then the value of $ \int\limits_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{{f(x)+f(-x)}{g(x)-g(-x)}dx} $ is
Options:
A) $ f(x)g(x) $
B) $ f(x)+g(x) $
C) 0
D) None of theses
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Let $ \phi (x)={ f(x)+f(-x) }{ g(x)-g(-x) } $ $ \phi ,(-x)={ f(-x)+f(x) }{ g(-x)-g(x) } $ $ =-{ f(x)+f(-x) }{ g(x)-g(-x) }=-\phi (x) $
$ \therefore \varphi (x) $ is an odd function
$ \Rightarrow \int\limits_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{\phi (x)dx=0} $
BETA
