I understand you can do it without the u sub I did, but if let's say this is all that came to your mind at the time what did I do wrong in my work?
You need to put the whole integrand in terms of u. So (avoiding the question of the limits)
$\displaystyle \int \frac{x}{\sqrt{1 - x^2}}~dx = \int \frac{1/2}{\sqrt{1 - u}}~du$
You can't call u $\displaystyle \sqrt{x}$ when you do the u integral.