I have been working through the new book "Physics: A student companion" discussed here: viewtopic.php?f=18&t=4475

I've encountered two things I'm stuck on. Hopefully this will be better than some recent threads where people copy/paste a problem from their homework and expect an answer! I'm just trying to brush up on my physics.

1) At the end of page 20 the book is showing how to find the equation of motion for a raindrop forming around a nucleation center, accumulating mass as it falls. It assumes that the raindrop accumulates mass exponentially with distance: dm/dz = alpha * m.

After a few steps there is a need to integrate: dv / [ g/v - alpha*v ]. According to the text, "Integrating (test by substitution), the velocity is: (answer). How did they integrate that expression? I asked one person who suggested multiplying the top and bottom of the fraction by v and then using partial fractions, although that seems different than what the author did.

2) Page 24 features a derivation of the rocket motion equations. For the case of gravity: m(t) * dv/dt = v_naut * m_dot - m(t) * g. Using separation of variables, both sides of the resulting equation is integrated. The left side is simply dv integrated to yield v(t). The right side expression, culminating in dt, is: m_dot/(m_naut - m_dot * t) * v_naut - g. The book integrates the expression from 0 to t. My first thought was to simply take the indefinite integral. However, if you take the indefinite integral you seem to get a different answer for the integration (due, I think, to the u substitution I used). What is the difference between integrating the expression from 0 to t, and taking the indefinite integral?

Thank you for any help!