Question

If f(x) = f(a – x) and $\underset{0}{\overset{a}{\int }}xf\left(x\right)dx=k\underset{0}{\overset{a}{\int }}f\left(x\right)dx,$ then k = ________________.

Answer 1

Given:
f(x) = f(a – x) ...(1)
$\underset{0}{\overset{a}{\int }}xf\left(x\right)dx=k\underset{0}{\overset{a}{\int }}f\left(x\right)dx$ ...(2)

$$\begin{gathered} \mathrm{Let}I=\underset{0}{\overset{a}{\int }}xf\left(x\right)dx...\left(3\right) \\ \mathrm{We}\mathrm{know}, \\ \underset{0}{\overset{a}{\int }}f\left(x\right)dx=\underset{0}{\overset{a}{\int }}f\left(a-x\right)dx \\ \mathrm{Thus}, \\ I=\underset{0}{\overset{a}{\int }}\left(a-x\right)f\left(a-x\right)dx \\ \Rightarrow I=\underset{0}{\overset{a}{\int }}\left(a-x\right)f\left(x\right)dx\left(\mathrm{From}\left(1\right)\right) \\ \Rightarrow I=\underset{0}{\overset{a}{\int }}af\left(x\right)dx-\underset{0}{\overset{a}{\int }}xf\left(x\right)dx \\ \Rightarrow I=\underset{0}{\overset{a}{\int }}af\left(x\right)dx-I\left(\mathrm{From}\left(3\right)\right) \\ \Rightarrow 2I=\underset{0}{\overset{a}{\int }}af\left(x\right)dx \\ \Rightarrow I=\frac{1}{2}\underset{0}{\overset{a}{\int }}af\left(x\right)dx \\ \Rightarrow I=\frac{a}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx...\left(4\right) \\ \mathrm{From}\left(2\right)\mathrm{and}\left(4\right), \\ k\underset{0}{\overset{a}{\int }}f\left(x\right)dx=\frac{a}{2}\underset{0}{\overset{a}{\int }}f\left(x\right)dx \\ \Rightarrow k=\frac{a}{2} \end{gathered}$$​

​Hence, k = $\underline{\frac{a}{2}}$.