Question

${\int }_0^{\infty }f(x+\frac{1}{x}).\frac{\ln x}{x}dx$ is equal to

• A. 0
• B. 1
• C. $\frac{1}{2}$
• D. cannot be evaluated

Answer 1

The correct option is A 0

$I={\int }_0^{\infty }f(x+\frac{1}{x}).\frac{\ln x}{x}dx$
Let $x=e^t$
$\Rightarrow dx=e^{t}dt$
$I={\int }_{-\infty }^{\infty }f(e^t+e^{-t})tdt=0$ since, $f(e^t+e^{-t})t$ is an odd function.
Ans: 0