Question

If $f(x)$ and $g(x)$ are continuous functions satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then $\int\limits_0^a f(x) \,g(x) $ $dx$ is equal to :

• A. $\int\limits_0^a g(x) d x$
• B. $ \int\limits_0^a f(x) d x$
• C. $0$
• D. $f(a)g(a)$

Answer 1

Answer: (B)

$I=\int\limits_0^a f(a-x) g(a-x) d x=\int\limits_0^a f(x)[2-g(x)] d x$
$I=2 \int\limits_0^a f(x) d x-\int\limits_0^a f(x) g(x) d x $
$ \Rightarrow I=\int\limits_0^a f(x) d x$