I am trying to remember again the stuff I did about nonlinear differential equations.I have...

I am trying to remember again the stuff I did about nonlinear differential equations.
I have
$\dot{x}=\left(\begin{array}{c}{x}_1^2\\ -1\end{array}\right)$
I want to solve this nonlinear differential equation and I know that the solution is:
$x_1(t)=\frac{x_1(0)}{1-x_1(0)}$
$x_2(t)=-t+x_2(0)$
I understand how to arrive to the expression of $x_2(t)$ but not to the one of $x_1(t)$.
If I integrate ${\dot{x}}_1=x_1^2$ I get
$\underset{0}{\overset{t}{\int }}{\dot{x}}_1(\tau )d\tau =\underset{0}{\overset{t}{\int }}{x}_1^2(\tau )$
which should give
$x_1(t)-x_1(0)={\left[\frac{x_1^3}{3}\right]}_{\tau =0}^{\tau =t}$
which does not give: $x_1(t)=\frac{x_1(0)}{1-x_1(0)}$
Can you help me?