So there is this question in "Conceptual Physics: The High School Physics Program" I wanted to clarify the answer, so I sought Barletby to see how it solved the question:
Chapter 8, Question 34A: https://www.bartleby.com/solution-answer/chapter-8-problem-34a-conceptual-physics-the-high-school-physics-program-9th-edition/9780133647495/in-the-previous-chapter-rocket-propulsion-was-explained-in-terms-of-newtons-third-law-that-is-the/98a06f98-9fd1-4c6a-9129-b7c593a7d40c

Can someone please explain to me how the problem was solved step by step, and why each part was done? I'd especially like to know why they differentiated and integrated the equation... what did they obtain by doing that?