Solve x = ∫ ve^-ktdt

$$x = \int v e ^ { - k t } d t$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{\ln(\frac{x-С}{tv})}{t}\text{, }&tv\neq 0\text{ and }x\neq С\text{ and }\left(v>0\text{ or }t<0\text{ or }x<С_{1}\right)\text{ and }\left(t<0\text{ or }x>С_{2}\text{ or }v<0\right)\text{ and }\left(v>0\text{ or }x>С_{3}\text{ or }t>0\right)\text{ and }\left(t>0\text{ or }v<0\text{ or }x<С_{4}\right)\\k\in \mathrm{R}\text{, }&x=С\text{ and }\left(t=0\text{ or }v=0\right)\end{matrix}\right.$$

Solve for v

$$\left\{\begin{matrix}v=\frac{xe^{kt}-Сe^{kt}}{t}\text{, }&t\neq 0\text{ and }k\neq 0\\v\in \mathrm{R}\text{, }&x=С\text{ and }t=0\text{ and }k\neq 0\end{matrix}\right.$$