Integration by substitution

I was checking my work against the mathematica integral calculator, and my answer on this problem is totaly different than theirs. Can anyone tell me what I did wrong?

$\displaystyle \int\frac{xdx}{\sqrt{1-x^{2}}}$

I did

$\displaystyle \mbox{Let }u=1-x^{2}, du=-2xdx $

$\displaystyle \int\frac{xdx}{\sqrt{1-x^{2}}}=\frac{-1}{2}\int(\sqrt{u})^{-1}du$

$\displaystyle \int\frac{xdx}{\sqrt{1-x^{2}}}=\frac{-1}{2}\ln|\sqrt{1-x^{2}}|+c$

mathmatica gives an answer of

$\displaystyle -\sqrt{1-x^{2}}$

I tries using $\displaystyle u=\sqrt{1-x^{2}} $, but that did not produce an equation I could integrate. Does anyone have any suggestions?