Answer
$\textbf v(t) = (e^t +1) \textbf i + (8t+3) \textbf j + \textbf k \\
\textbf r(t) = (e^t+t-1)\textbf i + (4t^2+3t) \textbf j + t\textbf k\\
\textbf r(2)= (e^2+1)\textbf i + 20 \textbf j + 2\textbf k $
Work Step by Step
$\textbf a(t) = e^t \textbf i + 8 \textbf k\\
\Rightarrow \textbf v(t) = \int (e^t \textbf i + 8 \textbf k) dt\\
\Rightarrow \textbf v(t) = e^t \textbf i + 8t\textbf j + \textbf C (t)\\$
According to the question, $\textbf v(0)= 2\textbf i + 3 \textbf j + \textbf k\\
\therefore \textbf i + \textbf C (t) = 2\textbf i + 3 \textbf j + \textbf k\\
\Rightarrow \textbf C (t) = \textbf i + 3 \textbf j + \textbf k \\
\therefore \textbf v(t) = (e^t +1) \textbf i + (8t+3) \textbf j + \textbf k\\
r(t) = \int \textbf v(t) dt\\
= (e^t+t)\textbf i + (4t^2+3t) \textbf j + t\textbf k+ \textbf C’(t) $
Now, according to the question, $r(0) = 0$
$\therefore 1\textbf i + \textbf C’ = 0\\
\Rightarrow \textbf C’ = -\textbf i $
Hence, $\textbf r(t) = (e^t+t-1)\textbf i + (4t^2+3t) \textbf j + t\textbf k \\
\textbf r(2)= (e^2+1)\textbf i + 20 \textbf j + 2\textbf k$
