Let $\{N_t\}$ be a homogeneous Poisson process with rate $\lambda>0$. I want to calculate the following conditional expectation: $$ E\left(\int_{t}^{1}\frac{1}{N_s}ds \;\Bigg|\;N_t\right),\;\;t\in(0,1). $$
It seems to me that $N_s\approx\lambda s$, so I conjectured that the integral should be $\frac{-\log t}{\lambda}$. But I am unable to prove that. Please help me or give me some hints. Thanks a lot.