Answer
$$\left\langle { - {e^{ - t}},{e^t},{t^3}} \right\rangle + {\bf{C}}$$
Work Step by Step
$$\eqalign{
& \int {\left\langle {{e^{ - t}},{e^t},3{t^2}} \right\rangle } dt \cr
& {\text{Integrating}} \cr
& \int {\left\langle {{e^{ - t}},{e^t},3{t^2}} \right\rangle } dt = \left\langle {\int {{e^{ - t}}dt} ,\int {{e^t}dt,\int {3{t^2}dt} } } \right\rangle \cr
& {\text{Integrating we obtain}} \cr
& \left\langle { - {e^{ - t}},{e^t},{t^3}} \right\rangle + {\bf{C}} \cr
& {\text{where }}{\bf{C}}{\text{ is a vector constant of integration}} \cr} $$
