Answer
\begin{align} r(t)&= \left\langle t+2,-t, 3 \right\rangle . \end{align}
Work Step by Step
By integration, we have
\begin{align} r(t)&=\int r'(t) dt\\
&=\int \left\langle1,-1,0 \right\rangle dt\\&= \left\langle t+c_1,-t+c_2,c_3 \right\rangle .
\end{align} By the condition $r(0)=\lt 2,0,3\gt$, we get $$2=c_1, \quad 0=c_2, \quad 3=c_3$$
Hence, we have \begin{align} r(t)&= \left\langle t+2,-t, 3 \right\rangle . \end{align}
