Hello,
how do we integrate (which is not that hard) but I can't remember the rule:
$$\frac{q}{m}\int E(t) dt $$ ?
What? The integral of E(t) depends entirely on what E(t) is! If E(t) is, say, $e^t$, then $\frac{q}{m}\int E(t)dt= \frac{q}{m}\int e^t dt= \frac{q}{m}e^t+ C$. If, however, $E(t)= e^{t^2}$, then $\frac{q}{m}\int E(t)dt= \frac{q}{m}\int e^{t^2}$ cannot be written in terms of any "elementary" function. (It can be written in terms of the "Error function" which is defined as $Erf(x)= \int_0^x e^{-t^2}dt$.