Question

$\underset{x\to \infty }{lim}\left(x^3{\int }_{1/x}^{1/x}\frac{ln(1+t^2)}{1+e^t}dt\right)$ equals

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

Given,

$\underset{x\to \infty }{lim}(x^3\cdot {\int }_{\frac{1}{x}}^{\frac{1}{x}}\frac{\ln (1+t^2)}{1+e^t}dt)$

${\int }_{\frac{1}{x}}^{\frac{1}{x}}\frac{\ln (1+t^2)}{1+e^t}dt=0$

as ${\int }_a^{a}f\left(x\right)dx=0,a\ne \infty$

$=\underset{x\to \infty }{lim}(x^3\cdot 0)$

$=0$