Velocity is the derivative of position with respect to time.
Let's call position vector r (don't want to use x because it is in x and z directions) so v = dr/dt
You can rewrite v = dr/dt as vdt = dr and then integrate both sides: ∫dr = ∫vdt
Let's substitute your function of velocity in to the integral: ∫dr = ∫(t i + t2 k) dt (remember i and k are just directions for the vector.)
r = 1/2t2 i + 1/3t3 k + C For t = 0, the particle is at 0i and 0k - this helps us solve for the constant.
0 = 1/2(0)2 i + 1/3(0)3 k + C so C = 0.