Question

A function f(x) satisfies $f\left(x\right)=f\left(\frac{c}{x}\right)$ for some real number $c(c>1)$ and $\forall x>0$. If $\underset{1}{\overset{\sqrt{c}}{\int }}\frac{f\left(x\right)}{x}dx=3$ then value of $\underset{1}{\overset{c}{\int }}\frac{f\left(x\right)}{x}dx$ is -

Answer 1

Put $x=\frac{c}{u}\Rightarrow dx=-\frac{c}{u^2}du$

$\underset{1}{\overset{\sqrt{c}}{\int }}\frac{f\left(x\right)}{x}dx=\underset{c}{\overset{\sqrt{c}}{\int }}\frac{uf\left(u\right)}{c}.(-\frac{c}{u^2})du=\underset{\sqrt{c}}{\overset{c}{\int }}\frac{f\left(u\right)}{u}du=3$

$\underset{1}{\overset{c}{\int }}\frac{f\left(x\right)}{x}dx=\underset{1}{\overset{\sqrt{c}}{\int }}\frac{f\left(x\right)}{x}dx+\underset{\sqrt{c}}{\overset{c}{\int }}\frac{f\left(x\right)}{x}dx=6$