Answer
$r(t)=(\frac{t^2}{2}+1)i+e^tj+(te^t-e^t +2)k$
or $r(t)=\lt \frac{t^2}{2}+1,e^t,te^t-e^t +2 \gt$
Work Step by Step
Consider $I=\int (ti+e^tj+te^t k)dt$
$I=Pi+ Qj+Rk$ ... (a)
$P=2t=\frac{1}{2}t^2+C$
$Q=e^t=e^t+C'$
and $R=te^t=te^t-\int e^t(1) dt=e^t(t-1) C''$
From equation (a):
$I=(\frac{1}{2}t^2+C)i+(e^t+C')j+(e^t(t-1) +C'')k$;(C,C',C'' are the constant of integration.)
From question, $r(0)=i+j+k$ gives $\lt 1,1,1 \gt$.
$r(0)=(\frac{1}{2}t^2+C)i+(e^t+C')j+(e^t(t-1)+ C'')k=j-k$
The desired result is:
$r(t)=(\frac{t^2}{2}+1)i+e^tj+(te^t-e^t +2)k$
or $r(t)=\lt \frac{t^2}{2}+1,e^t,te^t-e^t +2 \gt$
