Question

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

• A. is equal to zero
• B. is equal to one
• C. is equal to $\frac{1}{2}$
• D. can not be evaluated

Answer 1

Answer: (A)

is equal to zero

$I={\int }_0^{\infty }f(x+\frac{1}{x})\frac{\ln x}{x}dx$

Substitute $x=\frac{1}{t}\Rightarrow dx=-\frac{1}{t^2}dt$

$I={\int }_{\infty }^{0}f(t+\frac{1}{t})\frac{\ln \left(1/t\right)}{1/t}.\frac{-dt}{t^2}$

$={\int }_{\infty }^{0}f(t+\frac{1}{t})\frac{\ln t}{t}dt$

$={\int }_{\infty }^{0}f(x+\frac{1}{x})\frac{\ln x}{x}dx=-I$

$\Rightarrow 2I=0\Rightarrow I=0$