I think your mistake is when you say that X>1. It should be X>0.
The question asks to find the mean and variance of X = (1-Z)^(-1/2) where Z is randomly selected from [0,1].
Also recommends using a formula
I found the pdf for x= [1,infinity) and cdf
Calculating from here gives me an answer of 2. While the previous formula that uses CDF gives a different answer. Also I'm not sure how to find the second moment E(x^2), because it does not converge.
Any input on what I could be doing wrong is appreciated.