For context, the question I am attempting is the following from stochastic calculus: Show whether the stochastic integral is well-defined: $$\int_0^1 |W(t)|^{-1/2} dW(t)$$
where W(t) is standard Brownian Motion. I understand that I have to show it is adapted to a filtration F of Brownian motion, which I've done, and then also that it is square-integrable.
For square-integrability, I got to the integral of the expected value of the function squared equalling: $$2E[|Z|^{-1}]$$ but from here, am not sure how to show that this expectation (i.e. its integral) diverges. The solution tells me the integral is divergent because there is a non-integrable singularity at z=0, however i'm unsure how to show/determine this. I think I'm missing some background knowledge on how to compute expectations of standard normal variables. Any help would be appreciated!