Question

If $f\left(x\right)$ and $g\left(x\right)$ be continuous functions over the closed interval $[0,a]$ such that $f\left(x\right)=f(a-x)$ and $g\left(x\right)+g(a-x)=2.$ Then ${\int }_0^{a}f\left(x\right)\dot{g}\left(x\right)dx$ is equal to

• A. ${\int }_0^{a}f\left(x\right)dx$
• B. ${\int }_0^{a}g\left(x\right)dx$
• C. $2a$
• D. none of these

Answer 1

Answer: (A)

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

$I={\int }_0^{a}f\left(x\right).g\left(x\right)dx={\int }_0^{a}f(a-x).g(a-x)dx={\int }_0^{a}f\left(x\right).[2-g(x\left)\right]dx.....(\because f(a-x)=f(x\left)andg\right(a-x)=2-g(x\left)\right)={\int }_0^{a}2f\left(x\right)dx-{\int }_0^{a}f\left(x\right).g\left(x\right)dx={\int }_0^{a}2f\left(x\right)dx-I$

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

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