VC2.1

$\int f(t)\,dt=\int {(t-t^2)i+2t^3j-3k}\,dt\,$

= $i\int (t-t^2)\,dt+2j\int t^3\,dt-3k\int \,dt\,$

= $i[\frac{t^2}{2}-\frac{t^3}{3}]+\frac{t^4}{2}j-3tk+c\,$ where c is an arbitrary constant vector.

ii). $\int_1^2 f(t)\,dt=\int_1^2{(t-t^2)i+2t^3j-3k}\,dt\,$

= $i\int_1^2 (t-t^2)\,dt+2j\int_1^2 t^3\,dt-3k\int_1^2 \,dt\,$

= $i[\frac{t^2}{2}-\frac{t^3}{3}]+2j[\frac{t^4}{4}]-3k[t]\,$ at [1,2]

= $-\frac{5}{6}i+\frac{15}{2}j-3k\,$ (After Substituting the lower and upper values)