Your notation is very confusing.
1. What is v? Velocity? Does it depend on time? As in, is dv/dt = a?
2. What does v(ex) mean?
3. What does F^ext mean?
4. What is Mo?
5. I note that the answer you posted does not contain t. I would not expect that of an indefinite integration. Are you doing a definite integral? If so, what are the limits?