Let $B_t$ be a Brownian Motion starting at 0. Let $\tau_{-1}$ be the hitting time of $-1$, i.e. $\tau_{-1}=inf\{t\geq 0, B_t=-1\}$. How should we define the following integrals:
(1). $\overline{\lim_{t\rightarrow \tau_{-1}}} \int_{0}^t \frac{1}{(1+B_s)^2} ds$
(2). $\lim_{t\rightarrow \tau_{-1}} \int_0^t\frac{1}{(1+B_s)^2}dB_s$
Intuitively (1) should diverge, but how could we make it rigorous? I was thinking about using local time or law of iterated logarithm, but I did not see a way to write it out formally.
Thanks!