Limit to infinity of Lebesgue-Stieltjes integral with Brownian Motion

Question

Prove that for every fixed $x\in \mathbb{R}$

$$ \mathbb{E}_x \left[ \exp\big(\int_0^t \frac{1}{1+B_s^2} ds \big) \right] $$

goes to $\infty$ as $t\to \infty$.

Answer 1

Hints: By Jensen's Inequality $E_xe^{\int_0^{t} \frac 1 {1+B_s^{2}} ds} \geq exp(E_x{\int_0^{t} \frac 1 {1+B_s^{2}} ds})$. So it is enough to show that $E_x{\int_0^{t} \frac 1 {1+B_s^{2}} ds} \to \infty$. We can interchange the integral and the expectation and write down $E_x \frac 1 {1+B_s^{2}}$ using Gaussian density. I will let you take over from here.