Question

If $f(x)$ and $g(x)$ are continuous functions satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=2$, then $\underset{0}{\overset{a}{\int }}f(x)g(x)$ $dx$ is equal to :

• A. $\underset{0}{\overset{a}{\int }}g(x)dx$
• B. $\underset{0}{\overset{a}{\int }}f(x)dx$
• C. $0$
• D. $f(a)g(a)$

Answer 1

Answer: (B)

$I=\underset{0}{\overset{a}{\int }}f(a-x)g(a-x)dx=\underset{0}{\overset{a}{\int }}f(x)[2-g(x)]dx$
$I=2\underset{0}{\overset{a}{\int }}f(x)dx-\underset{0}{\overset{a}{\int }}f(x)g(x)dx$
$\Rightarrow I=\underset{0}{\overset{a}{\int }}f(x)dx$