Answer
$$\mathbf{r}=\left(\frac{t^{4}}{4}+2 t^{2}+1\right) \mathbf{i}+\left(\frac{t^{2}}{2}+1\right) \mathbf{j}+\frac{2 t^{3}}{3} \mathbf{k} $$
Work Step by Step
Since
\begin{align*}
\mathbf{r}&=\int\left[\left(t^{3}+4 t\right) \mathbf{i}+t \mathbf{j}+2 t^{2} \mathbf{k}\right] d t\\
&=\left(\frac{t^{4}}{4}+2 t^{2}\right) \mathbf{i}+\frac{t^{2}}{2} \mathbf{j}+\frac{2 t^{3}}{3} \mathbf{k}+\mathbf{C}
\end{align*}
\begin{align*}
\mathbf{r}(0)&=\left(\frac{0^{4}}{4}+2(0)^{2}\right) \mathbf{i}+\frac{0^{2}}{2} \mathbf{j}+\frac{2(0)^{3}}{3} \mathbf{k}+\mathbf{C}\\
\mathbf{i}+\mathbf{j}&=\mathbf{C}=\mathbf{i}+\mathbf{j}
\end{align*}
Hence
$$\mathbf{r}=\left(\frac{t^{4}}{4}+2 t^{2}+1\right) \mathbf{i}+\left(\frac{t^{2}}{2}+1\right) \mathbf{j}+\frac{2 t^{3}}{3} \mathbf{k} $$
