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 \[x= \frac{1}{3-t}\]
.