Question

If $f(x)=f(a-x)$, then $\underset{0}{\overset{a}{\int }}xf\left(x\right)dx$ is equal to

• A. $\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
• B. $\frac{a^2}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
• C. $\frac{a}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
• D. $-\frac{a}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx$

Answer 1

Answer: (C)

Let $I=\underset{0}{\overset{a}{\int }}xf\left(x\right)dx\dots \left(i\right)$

$\Rightarrow I=\underset{0}{\overset{a}{\int }}\left(a-x\right)f\left(a-x\right)dx$

$I=\underset{0}{\overset{a}{\int }}\left(a-x\right)f\left(x\right)dx\left[\because f\left(x\right)=f\left(a-x\right)given\right]$

$\Rightarrow I=a\underset{0}{\overset{a}{\int }}f\left(x\right)dx-\underset{0}{\overset{a}{\int }}xf\left(x\right)dx\dots \left(ii\right)$

On adding Eqs. $\left(i\right)$ and $\left(ii\right)$, we get

$\Rightarrow 2I=a\underset{0}{\overset{a}{\int }}f\left(x\right)dx$

$\Rightarrow I=\frac{a}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx$