Question

Let $f$ and $g$ be continuous functions on $[0,a]$ such that $f\left(x\right)=f(a-x)$ and $g\left(x\right)+g(a-x)=4,$ then ${\int }_0^{a}f\left(x\right)g\left(x\right)dx$ is equal to :-

• A. $4{\int }_0^{a}f\left(x\right)dx$
• B. $2{\int }_0^{a}f\left(x\right)dx$
• C. $-3{\int }_0^{2}f\left(x\right)dx$
• D. ${\int }_0^{a}f\left(x\right)dx$

Answer 1

Answer: (B)

$2{\int }_0^{a}f\left(x\right)dx$

$I={\int }_0^{a}f\left(x\right)g\left(x\right)dx$
$I={\int }_0^{a}f(a-x)g(a-x)dx$
$I={\int }_0^{a}f\left(x\right)(4-g(x\left)\right)dx$
$I=4{\int }_0^{a}f\left(x\right)dx-I$
$\Rightarrow I=2{\int }_0^{a}f\left(x\right)dx$