Answer
$$\frac{1}{4} \mathbf{i}+7 \mathbf{j}+\frac{3}{2} \mathbf{k}$$
Work Step by Step
Given $$ \int_{0}^{1}\left(t^{3} \mathbf{i}+7 \mathbf{j}+(t+1) \mathbf{k}\right) d t $$
Then
\begin{align*}
\int_{0}^{1}\left(t^{3} \mathbf{i}+7 \mathbf{j}+(t+1) \mathbf{k}\right) d t
&= \frac{t}{4}\bigg|_{0}^{1} \mathbf{i}+7 t\bigg|_{0}^{1} \mathbf{j}+\left(\frac{t^{2}}{2}+t\right)\bigg|_{0}^{1} \mathbf{k}\\
&=\frac{1}{4} \mathbf{i}+7 \mathbf{j}+\frac{3}{2} \mathbf{k}
\end{align*}
