Finding an Expression For Displacement When Velocity Given In Terms of Displacement

Suppose the velocity of a body is given in terms of the displacement. How can we find an expression for the displacement?
If  
\[v=2x^2\]
  and  
\[x=1\]
  when  
\[t=2\]
  (in the appropriate units), then we can write
\[v= \frac{dx}{dt}=2x^2\]

Now separate variables.
\[\frac{dx}{x^2}=x^{-2}dx=dt\]

Now integrate in the usual way.
\[\int^x_1 x^{-2}dx = \int^t_2 dt\]

\[[- \frac{1}{x} ]^x_1 = t-2\]

\[- \frac{1}{x} - ( \frac{1}{1} )=t-2\]

Making  
\[x\]
  the subject gives  
\[x= \frac{1}{3-t}\]
.