Answer
$2\mathrm{i}-4\mathrm{j}+32\mathrm{k}$
Work Step by Step
Use the last boxed formula of this section:
$\displaystyle \int_{0}^{2}(t\mathrm{i}-t^{3}\mathrm{j}+3t^{5}\mathrm{k})dt=(\int_{0}^{2}tdt)\mathrm{i}-(\int_{0}^{2}t^{3}dt)\mathrm{j}+(\int_{0}^{2}3t^{5}dt)\mathrm{k}$
$=[\displaystyle \frac{1}{2}t^{2}]_{0}^{2}\mathrm{i}-[\frac{1}{4}t^{4}]_{0}^{2}\mathrm{j}+[\frac{1}{2}t^{6}]_{0}^{2}\mathrm{k}$
$=\displaystyle \frac{1}{2}(4-0)\mathrm{i}-\frac{1}{4}(16-0)\mathrm{j}+\frac{1}{2}(64-0)\mathrm{k}$
$=2\mathrm{i}-4\mathrm{j}+32\mathrm{k}$
