mixster

Rocket has mass N with mass of fuel F.
The burning of the fuel uses up B mass of fuel per second, producing T newtons of thrust.
a is the acceleration of the rocket.
t is the time in seconds since launch.

F = m*a && a = F / m
=> a = T / (N + F - B*t)
integrating a follows 1/x => log|x| format
d/dt[N + F - B*t] = -B
integral(a)dt = (T) * log|N + F - B*t| * (-B) + u
    (where u is the constant of integration - using u instead of c due to it being velocity related)
=> (-T/B) * log|N + F - B*t| + u

The fuel will run out after F/B seconds, so...

integral(a, from 0, to F/B) = final velocity of rocket
integral(a, 0, F/B) = [(-T/B) * log|N + F - B*(F/B)|] - [(-T/B) * log|N + F - B*0|]   ## B*(F/B) = F; F - F = 0
integral(a, 0, F/B) = -[(T/B) * log|N|] + [(T/B) * log|N + F|]
integral(a, 0, F/B) = (T/B)[log|N + F| - log|N|]
integral(a, 0, F/B) = (T/B)[log|(N + F) / N|]