Expected Value of random variable in terms of hazard function

Question

I am reading Krishna's Auction Theory. I am currently stuck on Appendix A where he writes,

Now I understand equation A2,

what I don't follow is that for a hazard rate defined as below:

he writes

I have tried for a long time but unable to get this.

Answer 1

Here $F$ is the CDF of non-negative random variable $X$ with support $[0,\omega]$, and $f$ denotes its PDF. \begin{eqnarray*}\mathbb{E}\left(\frac{1}{\lambda(X)}\right) & = & \int_0^\omega\left(\frac{1-F(x)}{f(x)}\right)f(x) dx \\ & = & \int_0^\omega (1-F(x))dx \\ & = & \int_0^\omega \int_x^\omega f(t) dt dx \\ & = & \int_0^\omega\int_0^tf(t) dxdt \\ &=& \int_0^\omega f(t)\int_0^t dx dt \\ &=& \int_0^\omega f(t)t dt \\ & = & \mathbb{E}(X)\end{eqnarray*}