Find $r(t)$ if $r^{\text{'}}(t)=7t^{6}i10t^{9}j\sqrt[\phantom{2}]{tk}$ and $r(1)=ij.$

tep $1$

Integrals of vector functions are obtained by integrating each component separately. Therefore, if $r^{\text{'}}(t)=7t^{6}i10t^{9}j\sqrt[\phantom{2}]{t}k,$ then

$r(t)=i{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}7t^{6}dtj{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}10t^{9}dtk{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{t}dt$

$=c$