function loves integral

Calculus Level 4

$a\quad function\quad f\quad satisfies\quad f\left( x \right) =f(c/x)\quad for\quad some\quad real\quad number\\ c(>1)\quad and\quad all\quad positive\quad number\quad x.\quad \\ if\quad \int _{ 1 }^{ \sqrt { c } }{ f\left( x \right) /x\quad dx } =3\quad ,then\quad \int _{ 1 }^{ c }{ f(x)/x\quad dx } \quad equals?$ please post a solution .

The answer is 6.

1 solution

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Jul 1, 2017

$\displaystyle\int_1^\sqrt{c}\frac{f(x)}{x}dx=\int_\sqrt{c}^c\frac{f(\frac{c}{u})}{u}du=\int_\sqrt{c}^c\frac{f(u)}{u}du=3\quad\quad\text{by }u=\frac1{x}\text{ and }c>1$

Anding the two integrals together $\displaystyle\int_1^\sqrt{c}\frac{f(x)}{x}dx+\int_\sqrt{c}^c\frac{f(u)}{u}du=\int_1^c\frac{f(x)}{x}dx=3+3=6$

function loves integral

The answer is 6.

1 solution

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