Question

If f(a – x) = x and $\underset{0}{\overset{a}{\int }}f\left(x\right)dx=k\underset{0}{\overset{\frac{a}{2}}{\int }}f\left(x\right)dx,$ then k = _____________.

Answer 1

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

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

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


​Hence, k = 2.