Answer
$$s = \frac{1}{2}\arcsin \left( {{t^2}} \right) - \frac{1}{2}$$
Work Step by Step
$$\eqalign{
& \frac{{ds}}{{dt}} = \frac{t}{{\sqrt {1 - {t^4}} }} \cr
& {\text{Separate the variables}} \cr
& ds = \frac{t}{{\sqrt {1 - {t^4}} }}dt \cr
& {\text{Integrate both sides}} \cr
& \int {ds} = \int {\frac{t}{{\sqrt {1 - {t^4}} }}} dt \cr
& s = \frac{1}{2}\int {\frac{{2t}}{{\sqrt {1 - {{\left( {{t^2}} \right)}^2}} }}} dt \cr
& s = \frac{1}{2}\arcsin \left( {{t^2}} \right) + C{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Use the initial condition }}\left( {0, - \frac{1}{2}} \right) \cr
& - \frac{1}{2} = \frac{1}{2}\arcsin \left( 0 \right) + C \cr
& C = - \frac{1}{2} \cr
& {\text{Substitute }}C{\text{ into }}\left( {\bf{1}} \right) \cr
& s = \frac{1}{2}\arcsin \left( {{t^2}} \right) - \frac{1}{2} \cr
& \cr
& {\text{Graph}} \cr} $$
