A function f(x) satisfies f ( x ) = f ( c x ) for some real numbe… - Vidyadip

Question

A function $f(x)$ satisfies $f\left( x \right) = f\left( {\frac{c}{x}} \right)$ for some real number $c\left( {c > 1} \right)$ and $\forall\, x > 0$. If $\int\limits_1^{\sqrt c } {\frac{{f\left( x \right)}}{x}} dx = 3$ , then the value of $\int\limits_1^c {\frac{{f\left( x \right)}}{x}} dx$ is

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Answer

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

$\int_1^{\sqrt e } {\frac{{f(x)}}{x}} dx = \int_e^{\sqrt e } {\frac{{uf(u)}}{c}} \left( { - \frac{{{x^2}}}{c}} \right)du = \int_{\sqrt e }^e f (u)du$

$\int_1^e {\frac{{f(x)}}{x}} dx = \int_1^{\sqrt e } {\frac{{f(x)}}{x}} dx + \int_{\sqrt e }^e {\frac{{f(u)}}{u}} dx = 6$

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